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Chapter 4: Problem 1

A basketball player shoots four free throws, and you write down the sequence of hits and misses. Write down the sample space for thinking of this whole thing as a random experiment. In another game, a basketball player shoots four free throws, and you write down the number of baskets she makes. Write down the sample space for this different random experiment.

Short Answer

Expert verified
First sample space: All sequences of H and M in 4 shots (16 sequences). Second sample space: Number of baskets made (0 to 4).

Step by step solution

01

Understanding the First Experiment

In the first experiment, we record the sequence of hits (H) and misses (M) for four free throws. For example, HHHH means all shots are hits, while MMMM means all shots are misses. Each throw can independently be a hit or a miss.
02

List All Possible Sequences

There are four free throws, so there are 2 options (hit or miss) for each, leading to a total of 2^4 = 16 possible sequences. The sample space is:HHHH, HHMM, HHHM, HHMH, HMHM, HMMH, HMMM, MMMM, MMHH, MHHM, MHHH, MMHM, MHMM, MMHM, MMMH, MHMH.
03

Understanding the Second Experiment

In the second experiment, only the number of baskets (hits) made out of four free throws is recorded. This means we count the number of hits, regardless of their order.
04

Determine Possible Number of Baskets

The player can make 0, 1, 2, 3, or 4 baskets out of the four throws. Hence, the sample space for this experiment is simply the set of possible numbers of baskets made.
05

List the Sample Space

The sample space for the number of baskets made is: 0, 1, 2, 3, 4.

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Key Concepts

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

sample space
In probability theory, the sample space of an experiment is a set that contains all possible outcomes. For example, when a basketball player shoots four free throws, the sample space includes every combination of hits (H) and misses (M). There are 4 throws, each with 2 possible outcomes, so the sample space has 2^4 = 16 sequences:
HHHH, HHMM, HHHM, HHMH, HMHM, HMMH, HMMM, MMMM, MMHH, MHHM, MHHH, MMHM, MHMM, MMHM, MMMH, MHMH.
In the second experiment, we only count the number of baskets out of 4, so the sample space includes 0, 1, 2, 3, or 4 baskets.
random experiment
A random experiment is any process that results in one of several possible outcomes in a way that cannot be predicted with certainty. The outcome is determined by chance.
For example, shooting a basketball is a random experiment because each shot can either be a hit or a miss.
Another instance is counting the number of baskets made in four free throws, another kind of random experiment that still depends on the player鈥檚 skill and the inherent randomness of the game.
combinatorics
Combinatorics is the branch of mathematics focusing on counting, arrangement, and combination of objects.
When we list the sequences of hits and misses for four basketball throws, we use combinatorics to determine there are 2^4 = 16 outcomes.
In another scenario, if we are counting the number of made baskets, combinatorics helps us understand the various ways 0, 1, 2, 3, or 4 hits can occur in any order.
independent events
Independent events are those whose outcomes do not affect each other. Each free throw in our exercise is an independent event.
Whether the player makes or misses one shot doesn鈥檛 change the probability of making or missing the next.
Understanding this helps us calculate the likelihood of different sequences in the sample space accurately.

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