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Chapter 11: Problem 75

Explain how to find and probabilities with dependent events. Give an example.

Short Answer

Expert verified
Dependent events affect each other's outcomes. Their probabilities are found by multiplying the probability of the first event by the conditional probability of the second, given the first. For instance, the probability of drawing two aces in a row from a deck of cards is \( \frac{4}{52} \cdot \frac{3}{51} \).

Step by step solution

01

Introduction to Dependent Events

Dependent events are events in which the outcome of one event affects the outcome of another. For example, drawing a card from a deck and then drawing another without replacing the first one. The result of the second event is affected by the result of the first event.
02

Understanding Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. If the event of interest is A and event B has already occurred, the conditional probability of A given B is usually written as \( P(A | B) \).
03

Formula for Dependent Events

The probability of two dependent events A and B happening, \( P(A \cap B) \), is found by multiplying the probability of A by the conditional probability of B, given A. This can be formulated as \( P(A \cap B) = P(A) \cdot P(B | A) \).
04

Example

Let's have a deck of 52 cards. The probability of drawing an ace (A) on the first attempt is \( P(A) = \frac{4}{52} \). If we draw another card without replacing the first one, the probability of drawing another ace (B) would be dependent on the first event. Hence, the conditional probability \( P(B | A) = \frac{3}{51} \). Therefore, the probability of drawing two aces in a row would be \( P(A \cap B) = P(A) \cdot P(B | A) = \frac{4}{52} \cdot \frac{3}{51} \).

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