Jerry Stackhouse of the National Basketball Association's...

Jerry Stackhouse of the National Basketball Association's Dallas Mavericks is the best free- throw shooter on the team, making 89$\%$ of his shots (ESPN website, July, $2008 ) .$ Assume that late in a basketball game, Jerry Stackhouse is fouled and is awarded two shots. $\begin{array}{l}{\text { a. What is the probability that he will make both shots? }} \\ {\text { b. What is the probability that he will make at least one shot? }} \\ {\text { c. What is the probability that he will miss both shots? }}\end{array}$ $\begin{array}{l}{\text { d. Late in a basketball game, a team often intentionally fouls an opposing player in }} \\ {\text { order to stop the game clock. The usual strategy is to intentionally foul the other team's }} \\ {\text { worst free-throw shooter. Assume that the Dallas Mavericks' center makes } 58 \% \text { of his }}\end{array}$ free-throw shots. Calculate the probabilities for the center as shown in parts (a), (b), (b), and $(\mathrm{c}),$ and show that intentionally fouling the Dallas Mavericks' center is a better strategy than intentionally fouling Jerry Stackhouse.

Jerry Stackhouse of the National Basketball Association's Dallas Mavericks is the best free-
throw shooter on the team, making 89$\%$ of his shots (ESPN website, July, $2008 ) .$ Assume
that late in a basketball game, Jerry Stackhouse is fouled and is awarded two shots.
$\begin{array}{l}{\text { a. What is the probability that he will make both shots? }} \\ {\text { b. What is the probability that he will make at least one shot? }} \\ {\text { c. What is the probability that he will miss both shots? }}\end{array}$
$\begin{array}{l}{\text { d. Late in a basketball game, a team often intentionally fouls an opposing player in }} \\ {\text { order to stop the game clock. The usual strategy is to intentionally foul the other team's }} \\ {\text { worst free-throw shooter. Assume that the Dallas Mavericks' center makes } 58 \% \text { of his }}\end{array}$
free-throw shots. Calculate the probabilities for the center as shown in parts (a), (b), (b),
and $(\mathrm{c}),$ and show that intentionally fouling the Dallas Mavericks' center is a better
strategy than intentionally fouling Jerry Stackhouse.

Key Concepts

Strategic Game Decision Making

This concept involves using probability calculations to make informed decisions in competitive situations, such as sports. By comparing the probabilities of success between different players (for instance, a better versus a worse free-throw shooter), a team can adjust its strategy—like choosing whom to intentionally foul—to maximize favorable outcomes during the game.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability that it does not occur. This principle is particularly useful when calculating probabilities such as 'making at least one shot', where it is often simpler to first compute the probability of the opposite event (making no shots) and then subtract it from one.

Independent Events

This concept involves events whose outcomes are not influenced by each other. In probability, when multiple events are independent, the probability of all events occurring is the product of their individual probabilities. This principle is used to calculate the chance of consecutive successful free-throws if one shot does not affect the outcome of the other.

Bernoulli Trials and Binomial Probability

A Bernoulli trial is an experiment or process that leads to a binary outcome: success or failure, with a fixed probability of success. When repeated a fixed number of times, these independent trials form a binomial experiment. This framework is used to determine the probabilities of making a certain number of free-throws out of a given number of attempts.

Transcript

00:02 Hey guys, my name's colin, and let's go ahead and jump right into this probability problem.

00:07 We're given that jerry stackhouse of the dallas mavericks shoots 89 % of his free throws.

00:13 In other words, he has a 0 .89 probability of making a free throw.

00:17 And that late in the game, he's going to the free throw line for what are undoubtedly two critical shots.

00:23 Now, we're looking for the probability that he makes both, the probability that he makes at least one, or the probability that he misses both.

00:31 So before we can start, let's go ahead and calculate a couple more critical things to know.

00:37 The first is the probability that he misses.

00:40 Because probabilities are always between zero and one, that's how i got 89 % of his shots correlating to that probability that he makes it of 0 .89.

00:49 We can go ahead and calculate the probability that he misses it knowing that all probabilities, the total of all probabilities sums to one.

00:56 In other words, since he either makes the shot or he misses the shot and there's no other options, we can calculate that the probability that he misses the shot is just one minus the probability he makes the shot, and that result gives us 0 .11.

01:11 And since each of these free throws is independent of one another, in other words, the outcome of the second free throw is not at all dependent on the outcome of the first free throw, we can go ahead and calculate these as independent probabilities.

01:24 So, let's go ahead and start by calculating part a, the probability that he makes both shots.

01:31 Now, we know the probability he makes one shot is 0 .89.

01:36 So if he makes, he has a 0 .89 probability of making the first shot, and he is a 0 .89 probability of making the second shot.

01:43 And to calculate that answer for part a, all we do is simply multiply those two numbers together, and we get our result, 0 .7921.

01:54 And that right there is the probability that jerry makes both free throws.

01:58 Since part d, looking ahead, we're asked to compare that to the dallas maverick center, i'm going to go ahead and correlate, put .79211 down there so that we can compare it later to our answer.

02:11 Now for part b, we're looking at the probability that he makes at least one shot.

02:17 So intuitively, we can see that that's going to be the probability that he makes at least that he makes one shot plus the probability that he makes two shots.

02:24 Well, we just calculated the probability that he makes two shots from above.

02:28 So that right there, our work is done, but now we've got to go ahead and calculate the probability that he makes one shot.

02:33 And there are actually two different ways we can get to this outcome.

02:38 The first is that we can look at the outcome where he misses the first shot and makes the second shot.

02:47 In other words, a probability of 0 .11 that he misses the first shot and a probability of 0 .89 that he makes the second shot.

02:56 Or we can look at the outcome where he makes the first shot 0 .89 probability and misses the second shot.

03:07 And just like we did for part a above, we're going to multiply those two numbers together.

03:12 And when you do that, you end up with 0 .0979.

03:17 But since either one of these outcomes can give us that he only makes one shot, in order to calculate the total probability that he only makes one shot, we actually add them together.

03:29 And so when you do that, you add 0 .0979 to 0 .0979 and you get 0 .1958 as your answer there.

03:42 And that right there is the probability that he makes one shot.

03:48 So in order to calculate the probability that he makes one plus the probability that he makes two to get the probability that he makes at least one, we just add our answer from part a where he makes both shots to our answer from part b where he only makes one shot.

04:08 And when you add those two numbers together, you get a total probability of 0 .9879 as the probability that he makes at least one shot.

04:21 And again, i'm going to go ahead and write that down here in the corner for comparisons later...

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