A and B are two events if P(A)=1 4 P(B 2 5 and P(A ∪ B)=1/2 find the value of P(A ∩ B))

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.

Inhaltsverzeichnis Show

What is P(A/B) Formula?
P(A/B) Formula
P(A/B) Formula Examples
How to Find P(A∩B) using P(A/B) Formula?
What is ∩ Symbol in P(A∩B) Formula?
What is P(A∩B) Formula?
1. What is the Probability of A and B?
2. What is the Probability of A or B?

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)

Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts

Book a Free Trial Class

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”

Watch the video for a few quick examples of how to find the Probability of A and B / A or B:

Probability of A or B (also A and B)

Watch this video on YouTube.

Can’t see the video? Click here.

You may want to read this article first: Dependent or Independent Event? How to Tell the Difference.

Probability of A and B.
Probability of A or B.

A Venn diagram intersection shows events a and b happening together.

1. What is the Probability of A and B?

The probability of A and B means that we want to know the probability of two events happening at the same time. There’s a couple of different formulas, depending on if you have dependent events or independent events.

Formula for the probability of A and B (independent events): p(A and B) = p(A) * p(B).

If the probability of one event doesn’t affect the other, you have an independent event. All you do is multiply the probability of one by the probability of another.

Examples

Example 1: The odds of you getting promoted this year are 1/4. The odds of you being audited by the IRS are about 1 in 118. What are the odds that you get promoted and you get audited by the IRS?

Solution:
Step 1: Multiply the two probabilities together: p(A and B) = p(A) * p(B) = 1/4 * 1/118 = 0.002.

That’s it!

Example 2: The odds of it raining today is 40%; the odds of you getting a hole in one in golf are 0.08%. What are your odds of it raining and you getting a hole in one?

Solution:
Step 1: Multiply the probability of A by the probability of B. p(A and B) = p(A) * p(B) = 0.4 * 0.0008 = 0.00032.

That’s it!

Formula for the probability of A and B (dependent events): p(A and B) = p(A) * p(B|A)

The formula is a little more complicated if your events are dependent, that is if the probability of one event effects another. In order to figure these probabilities out, you must find p(B|A), which is the conditional probability for the event.

Example question: You have 52 candidates for a committee. Four are persons aged 18 to 21. If you randomly select one person, and then (without replacing the first person’s name), randomly select a second person, what is the probability both people will be between 18 and 21 years old?

Solution:
Step 1: Figure out the probability of choosing an 18 to 21 year old on the first draw. As there are 52 possibilities, and 4 are aged 18 to 21, you have a 4/52 = 1/13 chance.

Step 2: Figure out p(B|A), which is the probability of the next event (choosing a second person aged 18 to 21) given that the first event in Step 1 has already happened.
There are 51 people left, and only 3 are aged 18 to 21 now, so the probability of choosing a young adult again is 3/51 = 1 / 17.

Step 3: Multiply your probabilities from Step 1(p(A)) and Step 2(p(B|A)) together:
p(A) * p(B|A) = 1/13 * 1/17 = 1/221.

Your odds of choosing two people aged 18 to 21 are 1 out of 221.

2. What is the Probability of A or B?

The probability of A or B depends on if you have mutually exclusive events (ones that cannot happen at the same time) or not.

If two events A and B are mutually exclusive, the events are called disjoint events. The probability of two disjoint events A or B happening is:

p(A or B) = p(A) + p(B).

Example question: What is the probability of choosing one card from a standard deck and getting either a Queen of Hearts or Ace of Hearts? Since you can’t get both cards with one draw, add the probabilities:
P(Queen of Hearts or Ace of Hearts) = p(Queen of Hearts) + p(Ace of Hearts) = 1/52 + 1/52 = 2/52.

If the events A and B are not mutually exclusive, the probability is:

(A or B) = p(A) + p(B) – p(A and B).

Example question: What is the probability that a card chosen from a standard deck will be a Jack or a heart?
Solution:

p(Jack) = 4/52
p(Heart) = 13/52
p(Jack of Hearts) = 1/52

So:
p(Jack or Heart) = p(Jack) + p(Heart) – p(Jack of Hearts) = 4/52 + 13/52 – 1/52 = 16/52.

References

Salkind, N. (2019). Statistics for People Who (Think They) Hate Statistics 7th Edition. SAGE.

---------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Comments? Need to post a correction? Please Contact Us.