Problem 62 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.