Question

Two stars of the LA Clippers and the Golden State Warriors are very different when it comes to making free throws. ESPN.com reports that DeAndre Jordan makes \(81 \%\) of his free throw shots while Kevin Durant makes \(62 \%\) of his free throws. \({ }^{5}\) Assume that the free throws are independent and that each player shoots two free throws during a team practice. a. What is the probability that DeAndre makes both of his free throws? b. What is the probability that Kevin makes exactly one of his two free throws? c. What is the probability that DeAndre makes both of his free throws and Kevin makes neither of his?

Short answer

b. What is the probability that Kevin makes exactly one of his two free throws? c. What is the probability that DeAndre makes both of his free throws and Kevin makes neither of his?

Step-by-step solution

Identify the probability of success for DeAndre

Use the given information in the problem to find the probability of DeAndre making a free throw, which is 81%.

Apply the multiplication rule for independent events

Since DeAndre's free throw successes are independent events, we can use the multiplication rule to find the probability of both events happening. In this case, simply multiply the probability of success for each of the two free throws: \((0.81)(0.81)\).

Calculate the probability

After multiplying the probabilities, we get the probability of DeAndre making both of his free throws: \((0.81)(0.81)=0.6561\). Answer for part a: The probability that DeAndre makes both of his free throws is \(0.6561\). #b. The probability that Kevin makes exactly one of his two free throws#

Identify the probability of success and failure for Kevin

Use the given information in the problem to find the probability of Kevin making a free throw, which is 62%. The probability of him missing a free throw is therefore 38%.

Apply the addition rule for independent events

When calculating the probability of Kevin making exactly one of his two free throws, we need to consider the two possible scenarios: making the first free throw and missing the second, and missing the first free throw and making the second. The addition rule is used to combine these scenarios: \((0.62)(0.38)+(0.38)(0.62)\).

Calculate the probability

After adding the probabilities, we get the probability of Kevin making exactly one of his two free throws: \((0.62)(0.38)+(0.38)(0.62)=0.2356+0.2356=0.4712\). Answer for part b: The probability that Kevin makes exactly one of his two free throws is \(0.4712\). #c. The probability that DeAndre makes both of his free throws and Kevin makes neither of his#

Identify the probability of failure for Kevin

For this part, we will use the probability of Kevin missing a free throw calculated in part b, which is 38%.

Apply the multiplication rule for independent events

Since DeAndre's successes and Kevin's failures are independent events, we can use the multiplication rule for multiple independent events. In this case, multiply the probability of DeAndre making both of his free throws (from part a) by the probability of Kevin missing both of his free throws: \((0.6561)(0.38)(0.38)\).

Calculate the probability

After multiplying the probabilities, we get the probability of DeAndre making both of his free throws and Kevin making neither of his: \((0.6561)(0.38)(0.38)=0.09531516\). Answer for part c: The probability that DeAndre makes both of his free throws and Kevin makes neither of his is approximately \(0.0953\).

Key concepts

Understanding Independent Events through Basketball

In basketball, free throw shots by individual players often serve as good examples of independent events. Independent events are actions that do not influence each other; the outcome of one event does not affect the outcome of another.

In our basketball example with DeAndre Jordan and Kevin Durant, assuming their free throw attempts are independent means that whether DeAndre makes his first shot does not change the probability of him making his second shot, and the same applies to Kevin Durant.

This concept is vital because it allows us to analyze probabilities separately for each occurrence. For instance, DeAndre Jordan making two consecutive free throws are two independent events because making the first free throw has no bearing on the likelihood of making the second.

Understanding this concept can help students predict outcomes of sequences of events in sports and can be applied to various real-world scenarios where multiple, non-connected events take place.

Applying the Multiplication Rule to Calculate Probabilities

The multiplication rule is a probability principle that we use to determine the likelihood of two or more independent events occurring together. This rule simply states that to find the joint probability of multiple independent events, you multiply the probabilities of the individual events.

Returning to the basketball example, when we want to know the probability of DeAndre Jordan making two free throws in a row, we apply this rule. We calculate this by multiplying the probability of DeAndre making one free throw (\(0.81\) or 81%) by itself, because he needs to make both shots: \(0.81\times0.81=0.6561\).

If a student applies this rule, they will be able to assess the chances of different players making consecutive successful shots during a game or practice, which can lead to a better understanding of game strategies and player reliability.

Utilizing the Addition Rule in Probability

The addition rule comes into play when we want to calculate the probability of either of two mutually exclusive events occurring. Mutually exclusive means the events cannot happen at the same time—for instance, a player can't make and miss a shot simultaneously.

In the context of our basketball scenario, this rule helps us determine the probability of Kevin Durant making exactly one of his free throws. To figure this out, we must consider both ways Kevin could make one shot: either he makes the first and misses the second, or he misses the first and makes the second. We then add the probabilities of these two scenarios: \(0.62\times0.38+0.38\times0.62=0.4712\).

By understanding and applying the addition rule adequately, students can analyze situations involving multiple possible outcomes of a single event or two distinct events that cannot happen together, such as making a decision under uncertain conditions.