Problem 62 - College Algebra

Question

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

Step-by-Step Solution

Answer

With independent events, the probability of both events occurring is the product of their individual probabilities. So in an example where we draw a red card and roll a 3 on a die, the combined probability would be 1/12.

Step 1: Understand Independent Events

The first step in solving any problem related to independent events is understanding what independent events are. Two events are said to be independent if the occurrence of one event doesn't affect the occurrence of the other. In other words, if you already know that one event has occurred, this does not change the probability that the second event will occur.

Step 2: Apply Product Rule for Independent Events

After understanding independent events, the next step is to apply the product rule for independent events. This rule states that for two independent events A and B, the probability of both A and B happening together, denoted as P(A and B), is equal to the product of the probabilities of A and B happening individually i.e, P(A) * P(B). Each probability should be between 0 and 1 inclusive.

Step 3: Practical Example

Let's take an example where A is the event 'Draw a red card from a standard deck of cards' with P(A) = 1/2, and B is the event 'Roll a 3 on a fair six-sided die' with P(B) = 1/6. Since drawing a card and rolling a die are independent events (one does not affect the other), applying the product rule gives P(A and B) = P(A) * P(B) = (1/2) * (1/6) = 1/12.