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Chapter 8: Problem 62

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is not less than \(5 ?\)

Use the formula for \(_{n} C_{r}\) to solve Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 10-5+\frac{5}{2}-\frac{5}{4}+\dots-\frac{10}{1-\frac{1}{2}} $$

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