# Conditional probability problem

• toothpaste666
In summary, the probability of selecting a white ball from urn II is 19/32. The conditional probability of the transferred ball being white, given that a white ball is selected from urn II, is 9/19.
toothpaste666

## Homework Statement

Urn I contains 3 white and 5 red balls, whereas urn II contains 2 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II.

(a) What is the probability that the ball selected from urn II is white?

(b) What is the conditional probability that the transferred ball was white, given that a white ball is selected from urn II?

## The Attempt at a Solution

for part a) there are two cases. case 1 is that the ball taken from urn 1 was white and case 2 is that the ball taken from urn 1 was red.

case 1:
the first step is to pick the first ball (from urn 1). in this case it was white which has a 3/8 probability of happening
the next step is to pick the ball from urn 2. since the transferred ball was white, the probability of this is now 3/4 so in this case the probability that the first ball was white and the second ball was white is (3/8)(3/4) = 9/32

case 2:
the first step is picking a red ball from urn 1 which has a probability of 5/8. next step is to pick a white ball from urn 2 which since the transferred ball was red there is now a 2/4 probability of. So the probability that the first ball was red and the second ball was white is (5/8)(2/4) = 10/32

so the total probability that the ball from urn 2 is white is the sum of both case:
9/32 + 10/32 = 19/32

b) the probability that the ball taken from urn 1 was white given that the ball taken from urn 2 is white is
equal to (probability that the ball from urn 1 was white and the ball from urn 2 was white)/(probability that the ball from urn 2 was white) = (9/32)/(19/32) = 9/19

is my reasoning correct?

Yep.

toothpaste666
Thank you!

## What is conditional probability?

Conditional probability is a mathematical concept that measures the likelihood of an event occurring given that another event has already occurred. It takes into account the relationship between two events instead of just their individual probabilities.

## How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the intersection of two events by the probability of the first event. In other words, it is the probability of event A occurring given that event B has already occurred.

## What is the difference between conditional probability and regular probability?

The main difference between conditional probability and regular probability is that conditional probability takes into account a specific condition or event that has already occurred, while regular probability considers all possible outcomes without any additional information.

## What are some real-life examples of conditional probability?

Conditional probability is commonly used in fields such as medicine, economics, and statistics. For example, a doctor may use conditional probability to determine the likelihood of a patient having a certain disease based on their symptoms and medical history.

## How can conditional probability be used to solve problems?

Conditional probability can be used to solve problems by breaking down complex events into smaller, more manageable parts. By understanding the relationship between events, we can make more accurate predictions and decisions based on the available information.

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