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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. 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

Define the Possible Outcomes

We have two outcomes to analyze: whether the first free throw is made or missed, and if the second free throw is made or missed. This creates a 2x2 table with these categories: First made and second made, first made and second missed, first missed and second made, first missed and second missed.

02

Construct the Contingency Table

We populate the table with the given data:- First made, second made: 251- First made, second missed: 285 - 251 = 34- First missed, second made: 48- First missed, second missed: 53 - 48 = 5The contingency table will be:\[\begin{array}{c|c|c} & \text{Second Made} & \text{Second Missed} \hline\text{First Made} & 251 & 34 \hline\text{First Missed} & 48 & 5 \\end{array}\]

03

Calculate Total Attempts

To determine probabilities, calculate the total attempts: - Total 鈥渇irst made鈥� attempts = 251 + 34 = 285 - Total 鈥渇irst missed鈥� attempts = 48 + 5 = 53 - Total attempts = 285 + 53 = 338

04

Calculate the Requested Probabilities

To find the probability Bird made both free throws:- P(Both made) = \( \frac{251}{338} \)Calculate the conditional probability he made the second given he made the first:- P(Second made | First made) = \( \frac{251}{285} \)

05

Analyze Dependence

Compare the conditional probability to the overall probability of making the second free throw.- Overall P(Second made if made first) = \( \frac{299}{338} \)Since \( P(Second made | First made) \) is fairly close to the overall probability, it suggests his success on the second shot does not strongly depend on the first.

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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 measure of the likelihood that an event will occur. When computing probabilities, it's essential to consider the number of successful outcomes against the total number of possible outcomes.
In this exercise, Larry Bird's free throw attempts are examined to calculate several probabilities.
First, we start by looking at the total attempts.

• Larry made 251 out of 285 second free throws when he made his first.
• He made 48 out of 53 second throws after missing the first.
• Total second makes: 299 out of 338 attempts.

To calculate the probability that both free throws were made, we divide the number of successful pairs by the total: \[ P( \text{Both made} ) = \frac{251}{338} \approx 0.742 \] This probability suggests a strong likelihood that if he makes the first throw, he'll likely make the second.

Conditional Probability

Conditional probability represents how probabilities change when we know the outcome of a related event. It's calculated by considering the number of favorable outcomes compared to those that meet a certain condition.
In the context of Larry Bird鈥檚 free throws, we're interested in the probability that he made the second throw given that he made the first.
The calculation for conditional probability is: \[ P( \text{Second made} | \text{First made} ) = \frac{251}{285} \approx 0.881 \] This calculation tells us that around 88.1% of the time, Larry made the second shot after making the first.
This high probability suggests that if he starts successfully, he tends to continue being successful.

Independence in Probability

The concept of independence in probability explores whether one event affects the likelihood of another. In Larry Bird's case, we want to examine if making the first shot influences the second.
Mathematically, two events A and B are independent if: \[ P( B | A ) = P( B ) \] For this scenario, we compare the conditional probability of making the second shot given the first was made \( P(\text{Second made | First made}) = 0.881 \) to the overall probability of making any second shot \( P(\text{Second made}) = \frac{299}{338} \approx 0.884 \).
Since these probabilities are very close, it suggests that whether Larry makes the first shot does not strongly impact the probability of making the second. Therefore, for Larry Bird's free throws, making the first does not meaningfully alter the likelihood of making the second shot.