Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

If the probability of a basketball player's making a free throw is \(0.9\), find the probability that the player makes at least 1 of 2 free throws.

Step-by-Step Solution

Answer

The probability of making at least 1 of 2 free throws is 0.99.

Step 1: Understanding the Problem

We need to find the probability that the player makes at least one of two free throws. Let's denote by \( p \) the probability of making a free throw, which is given as \( p = 0.9 \). The complementary probability, or the probability that the player misses a free throw, is \( 1 - p = 0.1 \).

Step 2: Identifying the Complementary Event

The complementary event to making at least one free throw out of two is missing both free throws. If \( q = 0.1 \) is the probability of missing a free throw, then the probability of missing both free throws is \( q^2 = 0.1 \times 0.1 = 0.01 \).

Step 3: Calculating the Complementary Probability

Find the probability that both free throws are missed. Using the result from Step 2, the probability is 0.01.

Step 4: Using the Complementary Probability to Find the Desired Probability

The desired probability, \( P(\text{at least 1 made}) \), is the complement of both being missed: \[ P(\text{at least 1 made}) = 1 - P(\text{both missed}) = 1 - 0.01 = 0.99 \]

Step 5: Conclusion

Therefore, the probability that the player makes at least one of the two free throws is 0.99.

Key Concepts

Complementary Events in Probability Theory

In probability theory, a complementary event is essentially the opposite of the event you're interested in. For example, if you're looking at the possibility of an event happening, the complementary event is the possibility of it not happening. They always add up to 1. This is a key principle in probability. If you have the probability of an event, say making a basket, the probability of missing is its complement.

• If the probability of making a basket is 0.9, then the probability of missing it is 0.1.
• These two probabilities must sum to 1.

Complementary events are useful because sometimes it's easier to calculate the probability of something not happening and then subtract it from 1 to find the probability of the event occurring. In the context of our basketball example, after identifying the complementary event – missing both shots – we can easily determine the probability of making at least one shot by subtracting the probability of missing both from 1.

Understanding Compound Probability

Compound probability deals with the likelihood of two or more events occurring together. The events can be independent or dependent. In our scenario, the task is to find the probability of making at least one shot out of two attempts. This involves a sequence of events: making the first shot and missing the first but making the second, etc.

• To simplify, we identify the more straightforward complementary sequence – missing both shots.
• The probability of missing a single shot is known, as is making one.

By multiplying the probabilities of two dependent missed shots (0.1 each), we calculate the probability of missing both. Then, we use the complement of this outcome to find the total probability of making at least one shot. Compound probability allows us to handle more complex scenarios by breaking them down into sequence probabilities.

The Concept of Independent Events

Independent events are events whose outcomes do not affect each other. For example, flipping a coin twice, where the result of the first flip (heads or tails) doesn't change the probability of the second flip.

• In our basketball problem, each free throw is an independent event.
• The probability of making a free throw on the second attempt is not influenced by the result of the first attempt.

Independence is critical to our calculations because it means the probability calculations for each throw are straightforward. Whether the player makes or misses the first shot does not impact the second. Thus, when calculating the compound probability of two independent events, we simply multiply their probabilities.