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Chapter 5: Problem 33

Exercises 33 to 35 refer to the following setting. A basketball player makes 47% of her shots from the field during the season. To simulate whether a shot hits or misses, you would assign random digits as follows: (a) One digit simulates one shot; 4 and 7 are a hit; other digits are a miss. (b) One digit simulates one shot; odd digits are a hit and even digits are a miss. (c) Two digits simulate one shot; 00 to 47 are a hit and 48 to 99 are a miss. (d) Two digits simulate one shot; 00 to 46 are a hit and 47 to 99 are a miss. (e) Two digits simulate one shot; 00 to 45 are a hit and 46 to 99 are a miss.

Short Answer

Expert verified
Option (d) is the correct choice, as it accurately simulates a 47% shot success rate using two-digit numbers.

Step by step solution

01

Understand the Probability

The problem states that the probability of hitting a shot is 47%, which corresponds to a decimal probability of 0.47.
02

Evaluate Option (a)

In option (a), digits 4 and 7 represent a hit. There are 10 possible digits (0-9), so digits 4 and 7 constitute 2 out of 10, or 20% chance of a hit. This does not match the 47% probability.
03

Evaluate Option (b)

In option (b), all odd digits represent a hit (1, 3, 5, 7, 9). There are 5 odd digits, meaning a 50% chance of a hit, which is too high compared to 47%.
04

Evaluate Option (c)

In option (c), digits from 00 to 47 are a hit, covering 48 outcomes out of 100 (00 to 99). This results in a hit probability of 48%, slightly over our target of 47%.
05

Evaluate Option (d)

In option (d), digits from 00 to 46 are a hit, covering 47 outcomes out of 100. This results exactly in a 47% probability of hitting a shot.
06

Evaluate Option (e)

In option (e), digits from 00 to 45 are a hit, covering 46 outcomes out of 100. This produces a 46% probability of hitting a shot, under our target of 47%.
07

Conclusion

Compare all options to the required 47% probability. Options (c) and (d) are closest but (d) perfectly matches the 47% probability. Thus, option (d) is accurate.

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Key Concepts

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

Random Digits Assignment
When simulating real-world scenarios statistically, random digits can be a powerful tool. In this exercise, to simulate a basketball player's 47% shooting accuracy, we use random digits. First, we decide how many digits to use per shot. Options include using either one or two digits. Think of one digit as selecting a random number between 0-9, and two digits as selecting between 00-99. Then, assign these digits to represent hits or misses based on their probability.
For example, if using two digits, 00 to 46 could indicate a hit (47 digits, thus 47%), with 47 to 99 indicating a miss. This aligns exactly with the player's 47% shooting percentage. Choosing digits carefully ensures that the random simulation accurately mirrors reality, making random digits assignment an essential first step in probability simulation.
Probability Matching
The goal with random digit simulations is to match the probability of an actual event occurring鈥攊n this case, a player making a shot. This is called probability matching. Let's say a player makes 47% of their shots. In random digit simulation, this means we must allocate 47 out of 100 possible digit combinations to success.
In our exercise, option (d) is the perfect match for a 47% shooting accuracy鈥攚here digits 00 to 46 result in a successful shot. It's crucial that the simulation reflects the real-world probability as closely as possible for the results to be accurate. Without matching probabilities, any statistical analysis based on the simulation would be flawed.
Statistical Problem Solving
Statistical problem solving often involves creating models or simulations to understand real-world phenomena through numbers. This involves accurately identifying the probability, assigning values, and analyzing the outcomes.
In this basketball simulation, the statistical problem posed is to determine which method correctly reflects the player's 47% shooting success. By examining various digit assignment strategies, we solve the problem by finding option (d), which perfectly matches the required probability.
This step-by-step evaluation process is crucial in statistical problem solving, as it allows us to methodically test and verify which scenario best fits the given statistical data.

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Most popular questions from this chapter

Ten percent of U.S. households contain 5 or more people. You want to simulate choosing a household at random and recording whether or not it contains 5 or more people. Which of these are correct assignments of digits for this simulation? (a) Odd = Yes (5 or more people); Even = No (not 5 or more people) (b) 0 = Yes; 1, 2, 3, 4, 5, 6, 7, 8, 9 = No (c) 5 = Yes; 0, 1, 2, 3, 4, 6, 7, 8, 9 = No (d) All three are correct. (e) Choices (b) and (c) are correct, but (a) is not.

Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer. (a) Roll a die and record the count of spots on the up-face: \(P(1)=0, P(2)=1 / 6, P(3)=1 / 3, P(4)=1 / 3,\) \(P(5)=1 / 6, P(6)=0\) (b) Choose a college student at random and record gender and enrollment status: \(P(\text { female full-time })=\) \(0.56, P(\text { male full -time })=0.44, P(\text { female part-time })=\) \(0.24, P(\text { male part-time })=0.17\) (c) Deal a card from a shuffled deck: \(P(\text { clubs })=\) \(12 / 52, P\) (diamonds \()=12 / 52, P(\text { hearts })=12 / 52\) \(P(\text { spades })=16 / 52\) .

MySpace versus Facebook A recent survey suggests that 85% of college students have posted a profile on Facebook, 54% use MySpace regularly, and 42% do both. Suppose we select a college student at random. (a) Assuming that there are 20 million college students, make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected college student has posted a profile on at least one of these two sites. Write this event in symbolic form using the two events of interest that you chose in (b). (d) Find the probability of the event described in (c). Explain your method.

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Get rich A survey of 4826 randomly selected young adults (aged 19 to 25) asked, What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table shows the responses.14 Choose a survey respondent at random. (a) Given that the person selected is male, what鈥檚 the probability that he answered almost certain? (b) If the person selected said some chance but probably not, what鈥檚 the probability that the person is female?
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