Probability

We know that

Probability of occurrence of an event

Total outcomes of the coin are tails and heads

Hence the total no.of outcomes are 2 (i.e. heads and tails)

And the desired output is tail. Hence no.of desired outcomes = 1

Therefore, the probability of getting a tail is

Conclusion: Probability of getting a tail when a coin is flipped is 0.5 or

We know that,

Probability of occurrence of an event

Total outcomes are 1, 2, 3, 4, 5, 6, and the desired outcome is 5

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 1

Probability of getting 5

Conclusion: Probability of getting 5 when die is thrown is

We know that,

Probability of occurrence of an event

Total outcomes are 1, 2, 3, 4, 5, 6, and the desired outcomes are 2, 3

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 2

Probability of getting a 2 or 3

Conclusion: Probability of getting 2 or 3 when a die is thrown is

We know that,

Probability of occurrence of an event

As 1, 3, 5 are odd numbers up to 6, so the desired outcomes are 1, 3, 5, and total outcomes are 1, 2, 3, 4, 5, 6

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 3

Probability of getting an odd number

Conclusion: Probability of getting an odd number when a die is thrown is

We know that,

Probability of occurrence of an event

As 2, 3, 5 are prime numbers up to 6, so the desired outcomes are 2, 3, 5, and total outcomes are 1, 2, 3, 4, 5, 6

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 3

Probability of getting a prime number

Conclusion: Probability of getting a prime number when a die is thrown is

We know that,

Probability of occurrence of an event

As 3, 6 are multiples up to 6, so the desired outcomes are 3, 6, and total outcomes are 1, 2, 3, 4, 5, 6

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 2

Probability of getting multiple of 3

Conclusion: Probability of getting multiple of 3 when die is thrown is

We know that,

Probability of occurrence of an event

As 4, 5 are two numbers between 3 and, so the desired outcomes are 3, 6, and total outcomes are 1, 2, 3, 4, 5, 6

Therefore, total no.of outcomes are 6, and total no.of desired outcomes are 2

Probability of getting a number between 3 and 6

Conclusion: Probability of getting a number between 3 and 6 when a die is thrown is

(i) We know that,

Probability of occurrence of an event

Outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Total no.of outcomes are 36

In that only (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1) are our desired outputs as there sum is less than 6

Therefore no.of desired outcomes are 10

Therefore, the probability of getting a sum less than 6

Conclusion: Probability of getting a sum less than 6, when two dice are rolled is

(ii) We know that,

Probability of occurrence of an event

In (a, b) if a=b then it is called a doublet

Total doublets are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)

In (a, b) if a=b and if a, b both are odd then it is called a doublet

Odd doublets are (1, 1), (3, 3), (5, 5)

Outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Total no.of outcomes are 36 and desired outcomes are 3

Therefore, probability of getting doublet of odd numbers

Conclusion: Probability of getting doublet of odd numbers, when two dice are rolled is

(iii) We know that,

Probability of occurrence of an event

Outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Total no.of outcomes are 36

Desired outputs are (1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2), (5, 6), (6, 1), (6, 5)

Total no.of desired outputs are 15

Therefore, probability of getting the sum as a prime number

Conclusion: Probability of getting the sum as a prime number, when two dice are rolled is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are (1, 6), (3, 6), (5, 6)

Total no.of outcomes are 36 and desired outcomes are 3

Therefore, probability of getting odd on the first die and 6 on the second die

Conclusion: Probability of getting odd on the first die and 6 on the second die, when two dice are rolled is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are (4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)

Total no.of outcomes are 36 and desired outcomes are 9

Therefore, probability of getting number greater than 3 on each die

Conclusion: Probability of getting a number greater than 3 on each die, when two dice are rolled is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are (4, 6), (5, 5), (6, 4)

Total no.of outcomes are 36 and desired outcomes are 3

Therefore, the probability of getting a total of 10

Conclusion: Probability of getting total sum 10, when two dice are rolled is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are (3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)

Total no.of outcomes are 36 and desired outcomes are 10

Therefore, probability of getting total greater than 8

Conclusion: Probability of getting total greater than 8, when two dice are rolled is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are (3, 6), (4, 5), (5, 4), (6, 3), (6, 5) , (5, 6)

Total no.of outcomes are 36 and desired outcomes are 6

Therefore, probability of getting total equal to 9 or 11

Conclusion: Probability of getting total equal to 9 or 11, when two dice are rolled is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 9 balls is ⁹c₁ = 9

Our desired output is to pick a white ball. So, no.of ways to pick a white ball from 4 white balls (because the white ball can be picked from only white balls) is ⁴c₁ = 4

Therefore, the probability of picking a white ball

Conclusion: Probability of picking a white ball from 4 white balls and 5 white balls is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 20 balls is ²⁰c₁ = 20

Our desired output is to pick a red ball. So, no.of ways to pick a red ball from 9 red balls (because the red ball can be picked from only red balls) is ⁹c₁ = 9

Therefore, the probability of picking a red ball

Conclusion: Probability of picking a red ball from 9 red,

7 white and 4 black balls is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 20 balls is ²⁰c₁ = 20

Our desired output is to pick a white ball. So, no.of ways to pick a white ball from 7 white balls(because the white ball can be picked from only white balls) is ⁷c₁ = 7

Therefore, the probability of picking a white ball

Conclusion: Probability of picking a white ball from 9 red,

7 white and 4 black balls is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 20 balls is ²⁰c₁ = 20

Our desired output is to pick a white or red ball. So, no.of ways to pick a white or red ball from 16 balls(because there are a total of 16 balls which are either red or white) is ¹⁶c₁ = 16

Therefore, the probability of picking a white or red ball

Conclusion: Probability of picking a white or red ball from 9 red, 7 white, and 4 black balls is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 20 balls is ²⁰c₁ = 20

Our desired output is to pick a white or red ball. So, no.of ways to pick a white or red ball from 16 balls(because there are a total of 16 balls which are either red or white) is ¹⁶c₁ = 16

Therefore, the probability of picking a white or black ball

Conclusion: Probability of picking a white or black ball from 9 red, 7 white, and 4 black balls is

We know that,

Probability of occurrence of an event

By permutation and combination, total no.of ways to pick r objects from given n objects is ⁿC_r

Now, total no.of ways to pick a ball from 20 balls is ²⁰c₁ = 20

Our desired output is to pick a black or red ball(not white). So, no.of ways to pick a black or red ball from 13 balls (because there are a total of 13 balls which are either red or black) is ¹³c₁ = 13

Therefore, the probability of not picking a white ball

Conclusion: Probability of not picking a white ball from 9 red, 7 white, and 4 black balls is

We know that,

Probability of occurrence of an event

Total no.of outcomes = 10+25 = 35

Desired outcomes are prizes. Total no.of desired outcomes = 10

Therefore, the probability of getting a prize

Conclusion: Probability of getting a prize is

We know that,

Probability of occurrence of an event

Let B be Boy and G be Girl

Total possible outcomes are BB, BG, GB, GG

Our desired outcome is at least one boy. So, BB, BG, GB are desired outputs.

Total no.of outcomes are 4, and the desired outcomes are 3

Therefore, the probability of at least one boy

Conclusion: Probability of at least one boy is

We know that,

Probability of occurrence of an event

Let T be tails and H be heads

Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Desired outcomes are exactly two tails. So, desired outputs are TTH, THT, HTT

Total no.of outcomes are 8 and desired outcomes are 3

Therefore, the probability of getting exactly 2 tails

Conclusion: Probability of getting exactly two tails is

We know that,

Probability of occurrence of an event

Let T be tails and H be heads

Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Desired outcomes are exactly one tail. So, desired outputs are THH, HTH, HHT

Total no.of outcomes are 8 and desired outcomes are 3

Therefore, the probability of getting exactly one tail

Conclusion: Probability of getting exactly one tail is

We know that,

Probability of occurrence of an event

Let T be tails and H be heads

Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Desired outcomes are at most two tails. So, desired outputs are THH, HTH, HHT, TTH, THT, HTT, HHH

Total no.of outcomes are 8 and desired outcomes are 7

Therefore, the probability of getting at most 2 tails

Conclusion: Probability of getting at most two tails is

We know that,

Probability of occurrence of an event

Let T be tails and H be heads

Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Desired outcomes are at least two tails. So, the desired outputs are

TTH, THT, HTT, TTT

Total no.of outcomes are 8 and desired outcomes are 4

Therefore, the probability of getting at least 2 tails

Conclusion: Probability of getting at least two tails is

We know that,

Probability of occurrence of an event

Let T be tails and H be heads

Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Desired outcomes are at least two heads or at most two tails. So, desired outputs are TTH, THT, HTT, THH, HTH, HHT, HHH

Total no.of outcomes are 8 and desired outcomes are 7

Probability of getting at most 2 tails or at least 2 heads

Conclusion: Probability of getting at least two heads or at most two tails is

We know that,

Probability of occurrence of an event

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Desired outcomes are all outcomes except (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)

Total no.of outcomes are 36 and desired outcomes are 30

probability of not getting same number

Conclusion: Probability of not getting the same number on the two dice is

We know that,

Probability of occurrence of an event

Total possible outcomes are alphabets from a to z

Desired outcomes are a, e, i, o, u

Total no.of outcomes are 26 and desired outputs are 5

Therefore, the probability of picking a vowel

Conclusion: Probability of choosing a vowel is

(ii) We know that,

Probability of occurrence of an event

Total possible outcomes are all alphabets from a to z

Desired outcomes are all alphabets except a, e, i, o, u

Total no.of outcomes are 26 and desired outputs are 21

Therefore, the probability of picking a consonant

Conclusion: Probability of choosing a consonant is

We know that,

Probability of occurrence of an event

Total no.of outcomes are 52

Desired output is a number greater than 3 and less than 10.

There will be four sets of each card naming A, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. So, there will be a total of 24 cards between 3 and 10

Therefore, the probability of picking card between 3 and 10

Conclusion: Probability of picking a card between 3 and 10 is

We know that,

Probability of occurrence of an event

Total no.of outcomes are 12

Desired output is picking a number which is multiple of 2 or 3. So, desire outputs are 2, 3, 4, 6, 8, 9, 10, 12. Total no.of desired outputs are 8

Therefore, the probability of getting a number which is multiple of 2 or 3

Conclusion: Probability of picking a ticket which is multiple of 2 or 3 is

We know that,

Probability of occurrence of an event

An ordinary year has 365 days i.e. it has 52 weeks + 1 day. So, there will be 52 Tuesdays for sure(because every week has 1 Tuesday)

So, we want another Tuesday that to from that 1 day left(as there is only one Tuesday left after 52 weeks)
This one day can be, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.
Of these total 7 outcomes, the desired outcome is 1, i.e. Tuesday

Therefore, the probability of getting 52 Tuesdays in an ordinary year

Conclusion: Probability of getting 53 Tuesdays in an ordinary year is

We know that,

Probability of occurrence of an event

A leap has 366 days i.e. 52 weeks + 2 days. So, there will be 52 Sundays for sure (because every week has one Sunday)

So, we want another Sunday from the remaining two days.

The two days may be Sunday, Monday or Monday, Tuesday or Tuesday, Wednesday or Wednesday, Thursday or Thursday, Friday or Friday, Saturday or Saturday, Sunday

So, total outcomes are 7 and desired the outcomes are 2(Sunday, Monday or Saturday, Sunday)

Therefore, the probability of getting 53 Sundays in a leap year

Conclusion: Probability of getting 53 Sundays in a leap year is

We know that,

Probability of occurring = 1 - the probability of not occurring

Let’s calculate for the probability of not occurring, i.e. probability such that both of them don’t have a birthday on the same day. For suppose the first person has a birthday on a particular day then the other person can have a birthday in the remaining 364 days

Probability of not having the same birthday

Probability of having same birthday = 1 – probability of not having the same

Birthday

Conclusion: Probability of two persons having the same birthday is

(ii) and (iv) can’t be the probability of occurrence of an event. So, (ii) and (iv) are the answers to our question.

Explanation:

We know that,

i.e. probability can vary from 0 to 1(both are inclusive)

So, (i) 0 can be possible as

(ii) is not possible as it is less than 0

(iii) is possible as

(iv) is not possible as it is greater than 1

Conclusion: and are the probabilities that cannot occur.

We know that,

Probability of occurring = 1 - the probability of not occurring

Given the probability of occurrence

Therefore, the probability of not occurrence

Conclusion: Probability of not occurrence is

We know that,

If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is , which indirectly came from

Probability of the occurrence of an event

Where, Total no.of desired outcomes = a, and total no.of outcomes = a+b

Given a = 8, b= 13

The probability that the event occurs

Conclusion: Probability that the event occurs is

We know that,

If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is , similarly, if odds are not in the favor of the occurrence an event are a:b, then the probability of not occurrence of the event

is

We also know that,

Probability of occurring = 1 - the probability of not occurring

Given a = 4 and b = 7

Probability of occurrence

Conclusion: Probability that the event occurs is

(i) We know that,

If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is

Given, probability

We know, probability. So,

a = 5 and a+b = 14 i.e. b = 9

odds in favor of its occurrence = a:b

= 5:9

Conclusion: Odds in favor of its occurrence is 5:9

(ii) As we solved in part (i), a = 5 and b = 9

As we know, odds against its occurrence is b:a

= 9:5

Conclusion: Odds against its occurrence is 9:5

Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Total cases where sum will be 6 is (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) i.e. 5

Probability of getting sum 6

We know that, If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is

Now we got

So, a = 5 and a+b = 36 i.e. b = 31

Therefore odds in the favor of getting the sum as 6 is 5:31

Conclusion: Odds in favor of getting the sum as 6 is 5:31

(ii) Total outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) ,

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ,

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Total cases where sum will be 7 is (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) i.e. 6

Probability of getting sum 6

We know that,

If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is

Now we got

So, a = 1 and a+b = 6 i.e. b = 5

Therefore odds in the favor of getting the sum as 7 is 1:5

Odds against getting the sum as 7 is b:a i.e. 5:1

Conclusion: Odds against getting the sum as 7 is 5:1

As repetition is not allowed total no.of cases possible is

987(because if one of the numbers occupies a wheel, then the other wheel cannot be occupied by this number, i.e. next wheel have 1 less case than the previous wheel and so on)

Therefore, total cases = 504

Desired output is the correct combination of a single 3 digit number.

Therefore, the total no.of desired outcomes are 1

We know that,

Probability of occurrence of an event

Therefore, the probability of correct combination

Conclusion: Probability of guessing right combination is

All numbers are different (given in question), this will be the same as picking r different objects from n objects which is ⁿc_r

Here, n= 20 and r = 6(as we have to pick 6 different objects from 20 objects)

Now we shall calculate the value of ²⁰C₆ as ⁿC_r

i.e. ²⁰C₆ = 38760

Therefore, 38760 cases are possible, and in that only one them has prize, i.e. total no.of desired outcome is 1

As we know,

Probability of occurrence of an event

Therefore, the probability of winning a prize is

Conclusion: Probability of winning the prize in the game

is

Total no.of cases will be 6 x 6 x 6 = 216(because each die can have values from 1 to 6)

Desired outcomes are those whose sum up to 5. Desired outcomes are (1, 1, 3), (1, 3, 1), ( 1, 2, 2), (2, 1, 2), (2, 2, 1), (3, 1, 1) i.e. total of 6 cases

As we know,

Probability of occurrence of an event

Therefore, the probability of outcome whose sum is 5

Conclusion: Probability of getting a total of 5 when three dice are thrown is

(ii) Total no.of cases will be 6 x 6 x 6 = 216(because each die can have values from 1 to 6)

Desired outcomes are those whose sum up to 5. Desired outcomes are (1, 1, 3), (1, 3, 1), ( 1, 2, 2), (2, 1, 2), (2, 2, 1), (3, 1, 1) (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), i.e. total of 10 cases

As we know,

Probability of occurrence of an event

Therefore, the probability of outcome whose sum is at most 5

Conclusion: Probability of getting total of at most 5 when three dice are thrown is

Given : P(A) = 0.60, P(A or B) = 0.85 and P(A and B) = 0.42

To find : P(B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Substituting in the above formula we get,

0.85 = 0.60 + P(B) – 0.42

0.85 = 0.18 + P(B)

0.85 – 0.18 = P(B)

0.67 = P(B)

P(B) = 0.67

Given : P(A) = 0.4, P(A or B) = 0.6 and P(B) = 0.5

To find : P(A and B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Substituting in the above formula we get,

0.6 = 0.4 + 0.5 – P(A and B)

0.6 = 0.9 - P(A and B)

P(A and B) = 0.9 – 0.6

P(A and B) = 0.3

P(A and B) = 0.3

Given : P() = 0.4, P(A or B) = 0.7 and P(A and B) = 0.3

To find : P(B)

Formula used : P(A) = 1 – P()

P(A or B) = P(A) + P(B) - P(A and B)

We have P() = 0.4

P(A) = 1 – 0.4 = 0.6

We get P(A) = 0.6

Substituting in the above formula we get,

0.7 = 0.6 + P(B) – 0.3

0.7 = 0.3 + P(B)

0.7 – 0.3 = P(B)

0.4 = P(B)

P(B) = 0.4

_(i)

Given : P(A) = 0.25, P(A or B) = 0.5 and P(B) = 0.4

To find : P(A and B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Substituting in the above formula we get,

0.5 = 0.25 + 0.4 – P(A and B)

0.5 = 0.65 - P(A and B)

P(A and B) = 0.65 – 0.5

P(A and B) = 0.15

P(A and B) = 0.15

(ii) Given : P(A) = 0.25, P(A and B) = 0.15 ( from part (i))

To find : P(A and )

Formula used : P(A and ) = P(A) – P(A and B)

Substituting in the above formula we get,

P(A and ) = 0.25 – 0.15

P(A and ) = 0.10

P(A and ) = 0.10

_(i) Given : P(A) = 0.3, P(B) = 0.2 , P(A ∩ B) = 0.1

To find :

Formula used : = P(B) - P(A ∩ B)

Substituting in the above formula we get,

= 0.2 – 0.1

= 0.1

= 0.1

(ii) Given : P(A) = 0.3, P(B) = 0.2 , P(A ∩ B) = 0.1

To find :

Formula used : = P(A) - P(A ∩ B)

Substituting in the above formula we get,

= 0.3 – 0.1

= 0.2

= 0.2

Given : A and B are mutually exclusive events

P(A) = , P(B) =

To find : P(A or B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

For mutually exclusive events A and B, P(A and B) = 0

Substituting in the above formula we get,

P(A or B) = + – 0

P(A or B) =

P(A or B) =

Given : A and B are mutually exclusive events

P(not A) = P() = 0.65 , P(A or B) = 0.65

To find : P(B)

Formula used : P(A) = 1 – P()

P(A or B) = P(A) + P(B) - P(A and B)

For mutually exclusive events A and B, P(A and B) = 0

P(A) = 1 – P(not A)

P(A) = 1 – 0.65

P(A) = 0.35

Substituting in the above formula we get,

0.65 = 0.35 + P(B)

P(B) = 0.65 – 0.35

P(B) = 0.30

P(B) = 0.30

Given : A,B,C are mutually exclusive events and exhaustive events

P(B) = (3/2) P(A) and P(C) = (1/2) P(B)

To find : P(A)

Formula used : P(A) + P(B) + P(C) = 1

For mutually exclusive events A,B,and C , P(A and B) = P(B and C) = P(A and C)= 0

Let P(A) = x , P(B) = (3/2) P(A) = x , P(C) = (1/2) P(B) = x = x

x + x + x = 1

x = 1

x =

P(A) = x =

P(A) =

let A denote the event that a company executive will travel by plane and B denote the event of him travelling by train

Given : P(A) = , P(B) =

To find : Probability of a company executive will be travelling by plane or train=P(A or B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Probability of a company executive will be travelling in both plane and train =P(A and B)= 0

(as he cannot be travelling by plane and train at the same time)

P(A or B) = + – 0

P(A or B) =

P(A or B)=

Probability of a company executive will be travelling by plane or train= P(A or B) =

let A denote the event that the card drawn is king and B denote the event that card drawn is queen.

In a pack of 52 cards, there are 4 king cards and 4 queen cards

Given : P(A) = , P(B) =

To find : Probability that card drawn is king or queen = P(A or B)

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

P(A) = (as favourable number of outcomes = 4 and total number of outcomes = 52)

P(B) = (as favourable number of outcomes = 4 and total number of outcomes = 52)

Probability that card drawn is king or queen = P(A and B)= 0

(as a card cannot be both king and queen in the same time)

P(A or B) = + – 0

P(A or B) =

P(A or B)=

Probability of a card drawn is king or queen = P(A or B) =

let A denote the event that the card drawn is queen and B denote the event that card drawn is the heart.

In a pack of 52 cards, there are 4 queen cards and 13 heart cards

Given : P(A) = , P(B) =

To find : Probability that card drawn is either a queen or heart = P(A or B)

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

P(A) = (as favourable number of outcomes = 4 and total number of outcomes = 52)

P(B) = (as favourable number of outcomes = 13 and total number of outcomes = 52)

Probability that card drawn is both queen and heart = P(A and B)= 1

(as there is one card which is both queen and heart i.e queen of hearts)

P(A or B) = + – 1

P(A or B) =

P(A or B)=

Probability of a card drawn is either a queen or heart = P(A or B) =

let A denote the event that the card drawn is spade and B denote the event that card drawn is king.

In a pack of 52 cards, there are 13 spade cards and 4 king cards

Given : P(A) = , P(B) =

To find : Probability that card drawn is either a queen or heart = P(A or B)

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

P(A) = (as favourable number of outcomes = 13 and the total number of outcomes = 52)

P(B) = (as favourable number of outcomes = 4 and the total number of outcomes = 52)

The probability that card is drawn is both spade and king = P(A and B)= 1

(as there is one card which is both spade and king i.e. king of spades)

P(A or B) = + – 1

P(A or B) =

P(A or B)=

Probability of a card drawn is either a spade or king = P(A or B) =

let A denote the event that the number is divisible by 4 and B denote the event that the number is divisible by 4.

To find : Probability that the number is both divisible by 4 or 6 = P(A or B)

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

Numbers from 1 to 100 divisible by 4 are 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100.

There are 25 numbers from 1 to 100 divisible by 4

Favourable number of outcomes = 25

Total number of outcomes = 100 as there are 100 numbers from 1 to 100

P(A) =

Numbers from 1 to 100 divisible by 6 are

6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96

There are 16 numbers from 1 to 100 divisible by 6

Favourable number of outcomes = 16

Total number of outcomes = 100 as there are 100 numbers from 1 to 100

P(B) =

Numbers from 1 to 100 divisible by both 4 and 6 are

12,24,36,48,60,72,84,96

There are 8 numbers from 1 to 100 divisible by both 4 and 6

Favourable number of outcomes = 8

P(A and B) =

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) =

P(A or B) =

P(A or B)

The probability that the number is both divisible by 4 or 6 = P(A or B)

Given : A die is thrown twice

To find : Probability that at least one of the two throws comes up with the number 4

The formula used : Probability =

A die is numbered from 1 to 6

When a die is thrown twice, total number of outcomes = 6² = 36

Favourable outcomes =

{(4,1) ,(4,2) ,(4,3) ,(4,4) ,(4,5) ,(4,6) ,(1,4) ,(2,4) ,(3,4) ,(5,4) , (6,4)}

Favourable number of outcomes = 11

Probability that at least one of the two throws comes up with the number 4 =

The probability that at least one of the two throws comes up with the number 4 =

Given : two dice are tossed once

To find : Probability of getting an even number on the first die or a total 8.

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

A die is numbered from 1 to 6

When two dice are tossed once, total number of outcomes = 6² = 36

Let A denote the event of getting an even number on the first die and B denote the event of getting a total of 8

For getting an even number on the first die

Favourable outcomes =

{(2,1) ,(2,2) ,(2,3) ,(2,4) ,(2,5) ,(2,6) ,(4,1) ,(4,2) ,(4,3) ,(4,4) ,(4,5) ,(4,6) , (6,1) ,(6,2) ,(6,3) ,(6,4) ,(6,5) ,(6,6) }

Favourable number of outcomes = 18

Probability of getting an even number on the first die = P(A) =

For getting a total of 8

Favourable outcomes =

{ (2,6) , (4,4) , (6,2) ,(5,3) , (3,5) }

Favourable number of outcomes = 5

Probability of getting a total of 8 = P(A) =

For getting an even number on the first die and a total of 8

Favourable outcomes = {(2,6) , (4,4) , (6,2)}

Probability of getting an even number on the first die and a total of 8 = P(A and B) =

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) =

P(A or B) =

Probability of getting an even number on the first die or a total 8 =

Given: Two dice are thrown together.

Sample Space:

To Find: P(sum of faces neither divisible by 3 nor by 4)

Sum = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Sum neither divisible by 3 nor 4 = {2, 5, 7, 10, 11}

P(sum of faces neither divisible by 3 nor by 4)

Hence, probability is.

Given: Math students = 30%

Chemistry Students = 20%

Math & Chemistry both = 10%

To Find: P(Math or Chemistry)

Now, P(Math)

P(Chemistry)

P(Math ∩ Chemistry)

We know that,

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Therefore,

P(Math ∪ Chemistry) = 0.30 + 0.20 – 0.10 = 0.40

Hence, number of students studying math or chemistry are 40%.

let A denote the event that Hemant passes in english and B denote the event that hemant passes in hindi .

Given : P(A) = , P(B) = ,P(A and B) =

To find : Probability that he will pass in at least one of these subjects. = P(A or B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) =

P(A or B) =

P(A or B) =

The probability that he will pass in at least one of these subjects. = P(A or B) =

Let A denote the event that a person will get electrification contract and B denote the event that the person will get a plumbing contract

Given : P(A) = , P(not B) = P() = , P(A or B) =

To find: Probability that he will get both electrification and plumbing contract = P(A and B)

Formula used : P(B) = 1 – P()

P(A or B) = P(A) + P(B) - P(A and B)

P(B) = 1 -

P(B) =

Probability of getting at least one contract =

P(A and B)

P(A and B)

P(A and B)

P(A and B)

The probability that he will get both electrification and plumbing contract =

Let A denote the event that a patient visiting a denist will have a tooth extracted and B denote the event that a patient will have a cavity filled

Given : P(A) = 0.06 , P(B) = 0.2 , P(A or B) = 0.23

To find: Probability that he will have a tooth extracted and a cavity filled= P(A and B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Probability that he will have a tooth extracted or a cavity filled = 0.23

0.23 = 0.06 + 0.2 P(A and B)

0.23 = 0.26 P(A and B)

P(A and B) = 0.26 – 0.23 = 0.03

P(A and B) = 0.03

Probability that he will have a tooth extracted and a cavity filled=P(A and B) = 0.03

let A denote the event that the chosen individual is female and B denote the event that the chosen individual is over 50 years old.

Given : Town consists of 6000 people, 1200 are over 50 years old, and 2000 are females

To find : Probability that a randomly chosen individual from the town is either female or over 50 years = P(A or B)

The formula used : Probability =

P(A or B) = P(A) + P(B) - P(A and B)

For the event A ,

There are 2000 females present in a town of 6000 people

Favourable number of outcomes = 2000

Total number of outcomes = 6000

P(A) = =

For the event B,

There are 1200 are over 50 years of age in a town of 6000 people

Favourable number of outcomes = 1200

Total number of outcomes = 6000

P(A) = =

30% of the females are over 50 years

For the event A and B,

females are over 50 years of age

Favourable number of outcomes = 600

P(A and B) = =

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) =

P(A or B) =

P(A or B) =

The probability that a randomly chosen individual from the town is either female or over 50 years = P(A or B) =