# Conditional Probability with 3 Events

• MHB
• Math1015
In summary, the question involves conditional probability with 3 events and the task is to find the conditional probability of A given B, the complement of B, the probability of A, the conditional probability of C given A, and the odds in favor of B. You can use the formula P(A|B)= P(A∩B)/P(B) to solve this problem. The odds in favor of B are 2 out of 5 and the probability of B's complement is 1/3. With this understanding, you should be able to solve similar problems in the future.
Math1015
I'm currently stuck on a question that involves conditional probability with 3 events. This is a concept that I'm having the most trouble grasping and trying to solve in this subject. I am not sure how to start this problem.

The Question:
Given that P(A n B) = 0.4, P(A n C) = 0.2, P(B|A)=0.6 and P(B)=0.5, find the following.
a) P(A|B)
b) P (B')
c) P(A)
d) P (C|A)
e) the odds in favor of B

If someone could provide an explanation on how to solve this and guide me through it, that would be greatly appreciated! I really want to understand how to do problems like these.

$P(A|B)= \frac{P(A\cap B)}{P(B)}$. You are given that $P(A|B)= 0.6$ and that $P(A\cap B)= 0.4$ so $0.6= \frac{.4}{P(B)}$. $P(B)= \frac{0.4}{0.6}= \frac{2}{3}$. "(e) the odds in favor of B" are "2 out of 5" and "(b) P(B')= 1- 2/3= 1/3".

Thanks for the fast reply! We have an exam on monday and I've been stuck on this all day and understand it better now!

## 1. What is the definition of conditional probability with 3 events?

Conditional probability with 3 events is a mathematical concept that calculates the likelihood of an event occurring, given that two other events have already occurred. It is expressed as P(A|B,C) and represents the probability of event A happening, given that events B and C have already occurred.

## 2. How is conditional probability with 3 events calculated?

Conditional probability with 3 events is calculated using the formula P(A|B,C) = P(A and B and C) / P(B and C). This formula takes into account the intersection of all three events and the probability of the two conditional events occurring together.

## 3. Can the probability of an event change based on the occurrence of other events?

Yes, the probability of an event can change based on the occurrence of other events. This is the basis of conditional probability, as it takes into account the influence of other events on the likelihood of a specific event occurring.

## 4. How does conditional probability with 3 events differ from conditional probability with 2 events?

Conditional probability with 3 events is more complex than conditional probability with 2 events, as it takes into account the influence of two conditional events on the probability of the main event. It also involves calculating the intersection of all three events, rather than just the intersection of two events.

## 5. What are some real-world applications of conditional probability with 3 events?

Conditional probability with 3 events is commonly used in various fields such as medicine, finance, and statistics. It can be used to predict the risk of developing a disease based on a person's genetic and lifestyle factors, to determine the probability of a stock market investment being successful based on market trends and company performance, and to analyze data in surveys and experiments.

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