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Chapter 5: Problem 22

In a recent basketball game, a player who makes \(65 \%\) of his free throws made eight consecutive free throws. Assuming free-throw shots are independent, determine whether this feat was unusual.

Short Answer

Expert verified
Yes, the feat of making eight consecutive free throws is unusual, as it has a low probability of about 0.0302.

Step by step solution

01

- Understand the Problem

We need to determine if making eight consecutive free throws by a player with a 65% success rate is unusual. We will use probability concepts to solve this problem.
02

- Define the Probability of Success

The player makes 65% of his free throws, so the probability of making a single free throw is 0.65 or 65%.
03

- Define the Number of Trials

The player made eight free throws in a row. Thus, the number of trials (n) is 8.
04

- Calculate the Probability of Making All Free Throws

To find the probability of making eight consecutive free throws, we use the formula for independent events: \[ P(\text{all successes}) = P(\text{success})^n \]Thus, \[ P(\text{8 successes}) = 0.65^8 \]
05

- Compute the Result

Calculate the probability: \[ 0.65^8 \ \text{Using a calculator, we find} \ 0.65^8 \ \text{Approximately} \approx 0.0302 \]
06

- Interpret the Probability

A probability of about 0.0302 indicates that the event (making eight consecutive free throws) is quite unlikely. Any probability lower than 0.05 is usually considered unusual.

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Key Concepts

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

Independent Events
In probability theory, **independent events** are events where the occurrence of one event does not affect the occurrence of another. For example, in our basketball scenario, the player making a free throw is independent of all other attempts. This means each free throw success or miss does not affect the result of subsequent throws. Understanding this independence is crucial because it allows us to use simple probability rules to calculate complex outcomes. The formula for the probability of all independent events happening is the product of the probabilities of each individual event.
Probability Calculation
To calculate probability, we often use the formula: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
However, when dealing with independent events like the free throws in our exercise, we use a different formula: \( P(\text{all successes}) = P(\text{success})^n \).
This formula multiplies the probability of a single success by itself as many times as there are trials (free throws). For instance, with a 65% success rate in 8 consecutive throws, the calculation is \( 0.65^8 \), which gives an approximate value of 0.0302. This illustrates how probabilities compound when dealing with multiple trials.
Unusual Events
An event is considered **unusual** if it has a very low probability of occurring. A common threshold in statistics to categorize an event as unusual is a probability lower than 0.05 (or 5%).
In our given problem, making eight consecutive free throws with a success rate of 65% has a probability of about 0.0302, which is less than 0.05. Therefore, statistically, this streak is quite unlikely and can be termed unusual. Understanding this helps in evaluating the rarity or commonality of events occurring in real-life scenarios, especially in sports.

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Most popular questions from this chapter

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A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and wife, who are both carriers of the sickle- cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal-cell allele and one recessive sickle-cell allele. Therefore, the genotype of each parent is \(S s .\) Each parent contributes one allele to his or her offspring, with each allele being equally likely. (a) List the possible genotypes of their offspring. (b) What is the probability that the offspring will have sickle-cell anemia? In other words, what is the probability the offspring will have genotype \(s s ?\) Interpret this probability. (c) What is the probability that the offspring will not have sickle-cell anemia but will be a carrier? In other words, what is the probability that the offspring will have one dominant normal-cell allele and one recessive sickle- cell allele? Interpret this probability.

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