91Ó°ÊÓ

Go Celtics! Larry Bird, who played pro basketball for the Boston Celtics, was known for being a good shooter. In games during \(1980-1982,\) when he missed his first free throw, 48 out of 53 times he made the second one, and when he made his first free throw, 251 out of 285 times he made the second one. a. Form a contingency table that cross tabulates the outcome of the first free throw (made or missed) in the rows and the outcome of the second free throw (made or missed) in the columns. b. For a given pair of free throws, estimate the probability that Bird (i) made the first free throw and (ii) made the second free throw. (Hint: Use counts in the (i) row margin and (ii) column margin.) c. Estimate the probability that Bird made the second free throw, given that he made the first one. Does it seem as if his success on the second shot depends strongly, or hardly at all, on whether he made the first?

Step by step solution

01

Understand the Data

To solve the problem, we'll structure our information into a contingency table based on the outcomes of both the first and second free throws.

02

Create the Contingency Table

We are given that when Larry Bird missed his first free throw, he made the second one 48 times out of 53. Therefore, he missed the second one 5 times. Also, when he made his first free throw, he made the second one 251 times out of 285, missing it 34 times. We can create our contingency table as follows: \[\begin{array}{c|c|c} & \text{Made Second} & \text{Missed Second} \hline\text{Made First} & 251 & 34 \\text{Missed First} & 48 & 5 \\end{array}\]

03

Calculate Marginal Totals

Sum each row and column to create margins. For the first throws: 285 were made and 53 were missed. For the second throws: 299 were made and 39 were missed.Thus, we have: \[\begin{array}{c|c|c|c} & \text{Made Second} & \text{Missed Second} & \text{Total} \hline\text{Made First} & 251 & 34 & 285 \\text{Missed First} & 48 & 5 & 53 \hline\text{Total} & 299 & 39 & 338\end{array}\]

04

Calculate Probability of Making First and Second Free Throws

Calculate the probability that Bird made both the first and the second free throw: Since he made his first free throw 285 times, and out of those, he made the second 251 times, the probability is \( \frac{251}{338} \approx 0.742 \) or 74.2%.

05

Calculate Conditional Probability

Calculate the probability that Bird made the second free throw given he made the first: Divide the number of times he made the second throw after making the first (251) by the number of times he made the first throw (285). This gives \( \frac{251}{285} \approx 0.881 \) or 88.1%. There seems to be some dependency, although it appears he consistently performed well regardless of the outcome of the first throw.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation

Probability is the mathematical concept that measures the likelihood of an event happening. In this context, we calculate the probability by looking at how often an event happens out of the total number of events.
For example, to find the probability that Larry Bird made both the first and second free throws, we'll use the relevant data from the contingency table. We know he made his first throw 285 times and out of those, he made his second throw 251 times. So, the probability that he made both is calculated as the ratio of the successful consecutive attempts to the total attempts:

• Probability of making both: \( \frac{251}{338} \approx 0.742 \) or 74.2%

Understanding this probability helps in assessing Bird's consistent performance over the specified period.

Conditional Probability

Conditional probability asks how likely something is to happen given some other condition is already satisfied. In basketball terms, we want to know how likely Larry Bird is to make his second free throw if he's already made the first one. This is different from just looking at the chances of making any single throw as it considers the outcome of the first throw.

The calculation comes from only looking at the scenarios where Bird made his first throw and considering how many of these times he also made his second throw. We get this by using the formula:

• Probability he makes the second if he made the first: \( \frac{251}{285} \approx 0.881 \) or 88.1%

This result shows a dependency where making the first throw does seem to help with making the second, though Bird's overall high performance suggests he typically throws consistently well.

Marginal Totals

Marginal totals in a contingency table give you the overall numbers for each category of an event regardless of the outcome of another related event. Think of them like summaries that help you see the big picture of what's happening.

In the case of Larry Bird's free throws, we calculated:

• Total made first throws: 285
• Total missed first throws: 53
• Total made second throws: 299
• Total missed second throws: 39
• Overall total: 338 free throw pairs

These totals help provide context for understanding the conditional probabilities and the probabilities of each combination of throws. They form the basis for understanding the distribution of Bird's successes and misses, and thus, they guide us in predicting future performances.