Conditional Probability

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

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What IsP(A/B) Formula?

The conditional probability P(A/B) arises only in the case of dependent events. It givesthe conditional probability of Agiven that Bhas 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 sự kiện A happening.P(B) = Probability of event Bhappening.P(A∩B) =Probability of happening of both A và B.

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

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

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Note:If A và B are independent events, thenP(A/B) = P(A)or P(B/A) = P(B)



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Examples UsingP(A/B) Formula

Example 1:When a fair die is rolled, what is the probability of A given B where A is the sự kiện 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.

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Using theP(A/B) formula:

P(A/B) =P(A∩B) / P(B)

(P(A/B) = dfrac2/63/6 = dfrac 2 3)

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

Example 2:Two cards are drawn from a deông chồng of 52 cards where the first thẻ is NOT replaced before drawing the second thẻ. 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/thẻ 1 is a king) = 3 / 51 (as the first king is not replaced, there is a totalof 3 kings out of 51 left out cards).

Bythe formula of conditional probability,

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

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

Answer: The required probability= 1 / 221.

FAQs onP(A/B) Formula

What IsP(A/B) Formula?

P(A/B) Formula is the formula used to lớn calculate the conditional probability such that we have to lớn 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) isthe probability of event Bhappening andP(A∩B) is the probability of happening of both A và B.

How to FindP(A∩B) UsingP(A/B) Formula?

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

What Is ∩ Symbol inP(A∩B) Formula?

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