Question

During the \(2015-16\) NBA season, Stephen Curry of the Golden State Warriors had a free throw shooting percentage of 0.908 . Assume that the probability Stephen Curry makes any given free throw is fixed at 0.908 , and that free throws are independent. (a) If Stephen Curry shoots two free throws, what is the probability that he makes both of them? (b) If Stephen Curry shoots two free throws, what is the probability that he misses both of them? (c) If Stephen Curry shoots two free throws, what is the probability that he makes exactly one of them?

Short answer

The probability that Stephen Curry makes both of two free throws is approximately 0.825, the probability that he misses both is approximately 0.008, and the probability that he makes exactly one is approximately 0.167.

Step-by-step solution

Determine the probability of Stephen Curry making two free throws

The probability of Stephen Curry making a free throw is given as 0.908. Because each free throw is an independent event, the probability of him making two in a row is the product of the probabilities of each individual throw, i.e., \(0.908 * 0.908 = 0.825264\). So, the probability that he makes both free throws is approximately 0.825.

Determine the probability of Stephen Curry missing two free throws

The probability of Stephen missing a free throw is \(1 - 0.908 = 0.092\). Again, since these are independent events, the probability of him missing two in a row is the product of the probabilities of each individual throw, i.e., \(0.092 * 0.092 = 0.008464\). So, the probability that he misses both free throws is approximately 0.008.

Determine the probability of Stephen Curry making exactly one free throw

There are two possibilities for Stephen Curry making exactly one free throw: he can either make the first and miss the second or miss the first and make the second. The probability of either of these events happening is \(0.908 * 0.092\), which is approximately 0.083536. Since these are mutually exclusive events (he can't both make the first and miss it, and vice versa), we can add these probabilities together to get the overall probability. So, the probability that he makes exactly one free throw is approximately \(2 * 0.083536 = 0.167072\).

Key concepts

Independent Events

When it comes to probability theory, understanding independent events is crucial. Imagine flipping a coin: whether it lands heads or tails on one flip does not affect its outcome on the next flip; each flip is independent of the others. This is the essence of independent events. They are outcomes that are not influenced by previous events.

In the case of the NBA star Stephen Curry's free throws, each shot he takes is an independent event because the outcome of one shot—whether it's a make or a miss—doesn't affect the next shot. This independence is important when we calculate combined probabilities of multiple events, like predicting if he'll make two shots in a row. To answer such questions, we apply the probability multiplication rule.

Probability Multiplication Rule

The probability multiplication rule is a key principal in probability theory used to determine the likelihood of two or more independent events occurring together. The rule simply states that to find the probability of both events happening, you multiply the probabilities of each event occurring by itself.

For Stephen Curry's free throws, the multiplication rule is applied as follows: the probability of making the first free throw is 0.908, and since free throws are independent events, the probability he makes the second free throw is also 0.908. Therefore, the likelihood he makes both is calculated by multiplying these probabilities: \(0.908 \times 0.908\), which equals approximately 0.825.

• If he makes the first shot, the chance of making the second doesn't change.
• If he misses the first shot, this has no impact on the second shot—its chance remains the same.

This rule is essential for understanding the outcomes of a sequence of independent events.

Mutually Exclusive Events

Mutually exclusive events are another fundamental concept in probability theory. These are events that cannot both occur at the same time. Think of it as being impossible for Stephen Curry to both make and miss the same free throw - only one outcome can happen.

When calculating Curry's probability of making exactly one free throw out of two, we have two mutually exclusive scenarios: either he makes the first and misses the second, or misses the first and makes the second. The first scenario has a probability of \(0.908 \times 0.092\), and so does the second, since the outcomes are reversed. As these scenarios cannot occur at the same time, we add the two probabilities to get the final answer. This is where the formula for mutually exclusive events comes into play, suggesting the overall probability is the sum of individual probabilities: \(2 \times 0.908 \times 0.092\), resulting in approximately 0.167.

Key Points for Mutually Exclusive Events:

• Only one can occur at a given time.
• Overall probability is the sum, not the product, of individual probabilities.
• Applying this to multiple choices results in a combined probability covering all possible exclusive outcomes.