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Chapter 8: Problem 62

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

Short Answer

Expert verified
The probability of independent events occurring is the product of the probabilities of each event, given by the formula \( P(A \cap B) = P(A) * P(B) \). For example, the probability of flipping a heads and rolling a 6 is \( \frac{1}{12} \).

Step by step solution

01

Understanding Independent Events

Independent events in probability are events that the occurrence of one does not affect the occurrence of the other. For example, if you flip a coin and roll a die, the result of the coin flip does not affect the outcome of the roll of the die. These two events are independent.
02

Probability of Independent Events

The probability of independent events occurring is the product of the probabilities of each event. This is defined by the formula: \[ P(A \cap B) = P(A) * P(B) \] where \( A \) and \( B \) are the independent events, \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) happening, \( P(A) \) is the probability of event \( A \) happening, and \( P(B) \) is the probability of event \( B \) happening.
03

Example Calculation

Let's consider an example. You flip a coin (event A) and roll a die (event B). The probability of getting a heads (event A) is \( \frac{1}{2} \) and the probability of getting a 6 on the die roll (event B) is \( \frac{1}{6} \). Using our formula, the probability of both these independent events occurring is: \[ P(A \cap B) = P(A) * P(B) = \frac{1}{2} * \frac{1}{6} = \frac{1}{12} \] So, there is a \( \frac{1}{12} \) chance of flipping a heads and rolling a 6 at the same time.

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