If a and b are two events such that p(a∪b 2 3 and p(a 1 3 find the probability of))

P(A/B) is known as conditional probability and it means the probability of event A that depends on another event B. It is also known as "the probability of A given B". P(A/B) Formula is used to find this conditional probability quickly.

What is P(A/B) Formula?

The conditional probability P(A/B) arises only in the case of dependent events. It gives the conditional probability of A given that B has occurred.

P(A/B) Formula

P(A/B) = P(A∩B) / P(B)

Similarly, the P(B/A) formula is: P(B/A) = P(A∩B) / P(A)

Here,

P(A) = Probability of event A happening.

P(B) = Probability of event B happening.

P(A∩B) = Probability of happening of both A and B.

From these two formulas, we can derive the product formulas of probability.

• P(A∩B) = P(A/B) × P(B)
• P(A∩B) = P(B/A) × P(A)

Note: If A and B are independent events, then P(A/B) = P(A) or P(B/A) = P(B)

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P(A/B) Formula Examples

Example 1: When a fair die is rolled, what is the probability of A given B where A is the event of getting an odd number and B is the event of getting a number less than or equal to 3?

Solution:

To find: P(A/B) using the given information.

When a die is rolled, the sample space = {1, 2, 3, 4, 5, 6}.

A is the event of getting an odd number. So A = {1, 3, 5}.

B is the event of getting a number less than or equal to 3. So B = {1, 2, 3}.

Then A∩B = {1, 3}.

Using the P(A/B) formula:

P(A/B) = P(A∩B) / P(B)

\(P(A/B) = \dfrac{2/6}{3/6} = \dfrac 2 3\)

Answer: P(A/B) = 2 / 3.

Example 2: Two cards are drawn from a deck of 52 cards where the first card is NOT replaced before drawing the second card. What is the probability that both cards are kings?

Solution:

To find: The probability that both cards are kings.

P(card 1 is a king) = 4 / 52 (as there are 4 kings out of 52 cards).

P(card 2 is a king/card 1 is a king) = 3 / 51 (as the first king is not replaced, there is a total of 3 kings out of 51 left out cards).

By the formula of conditional probability,

P(card 1 is a king ∩ card 2 is a king) = P(card 2 is a king/card 1 is a king) × P(card 1 is a king)

P(card 1 is a king ∩ card 2 is a king) = 3 / 51 × 4 / 52 = 1 / 221

Answer: The required probability = 1 / 221.

P(A/B) Formula is the formula used to calculate the conditional probability such that we have to find the probability of event 'A' occurring when event 'B' has occurred. P(A/B) Formula is given as, P(A/B) = P(A∩B) / P(B), where, P(A) is probability of event A happening, P(B) is the probability of event B happening and P(A∩B) is the probability of happening of both A and B.

How to Find P(A∩B) using P(A/B) Formula?

P(A∩B) can be calculated using the P(A/B) Formula as, P(A∩B) = P(A/B) × P(B), where, P(B) is the probability of event B happening and P(A∩B) is the probability of happening of both A and B.

What is ∩ Symbol in P(A∩B) Formula?

P(A/B) Formula is given as, P(A/B) = P(A∩B) / P(B), here ∩ symbol represents the intersection of event 'A' and event 'B'. P(A) is probability of event A happening, P(B) is the probability of event B happening and P(A∩B) is the probability of happening of both A and B.

What is P(A∩B) Formula?

P(A∩B) is the probability of both independent events “A” and "B" happening together, P(A∩B) formula can be written as P(A∩B) = P(A) × P(B),
where,

• P(A∩B) = Probability of both independent events “A” and "B" happening together.
• P(A) = Probability of an event “A”
• P(B) = Probability of an event “B”

Text Solution

`P(A cap B) ge (2)/(3)``P(A cap overline(B))+ ge (1)/(3)``(1)/(6) le P(A cap B) le (1)/(2)``(1)/(6) le P(overline(A) cap B) le (1)/(2)`

Answer : B

Solution : We have, <br> `P(A cap B) ge " max" {P(A),P(B)}=2//3` <br> and, `P(A cap B) le " min" {P(A),P(B)}=1//2` <br> So, option (a) is correct. <br> Now, <br> `P(A cap B)=P(A)+P(B)-P(A cup B)` <br> `implies P(A cap B) ge P(A)+P(B)-1=(1)/(6)` <br> `therefore (1)/(6) le P(A cap B) le (1)/(2)` <br> So, option (c ) is correct. <br> Now, `(1)/(6) le P(A cap B) le (1)/(2)` <br> `implies -(1)/(2) le -(P cap B) le -(1)/(6)` <br> `implies (2)/(3)-(1)/(2) le P(B)-P(A cap B) le(2)/(3)-(1)/(6) le P(overline(A) cap B) le (1)/(2)` <br> So, option (d) is correct. <br> Now, `P(A cap overline(B))=P(A)-P(A cap B)` <br> `implies P(A cap overline(B)) le (1)/(2)-(1)/(6)=(1)/(3)` <br> So, option (d) is correct.