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Chapter 6: Problem 49

Movies sample Five of the 20 movies running in movie theatres this week are comedies. A selection of four movies are picked at random. Does \(X=\) the number of movies in the sample which are comedies have the binomial distribution with \(n=4\) and \(p=0.25 ?\) Explain why or why not.

Short Answer

Expert verified
No, the distribution of selecting movies is not binomial because the trials are not independent and the success probability changes.

Step by step solution

01

Understanding Binomial Distribution

The binomial distribution is applicable when there are a fixed number of trials, each with two possible outcomes (such as success and failure), and each trial is independent with the same probability of success. Here, the trials are selecting movies, the success condition is picking a comedy, and the failure condition is not picking a comedy.
02

Check Independent Trials

For the binomial distribution, each trial must be independent. In this case, selecting one movie influences the outcome of selecting the next because the total pool of movies decreases without replacement. Hence, the trials are not independent.
03

Check Fixed Success Probability

To fit a binomial model, the probability of success must remain constant across trials. Initially, the probability of selecting a comedy is \( p = \frac{5}{20} = 0.25 \). However, as movies are selected, this probability changes鈥攊f a comedy is picked first, fewer comedies remain, and if a non-comedy is picked, the chance of picking a comedy increases.

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Key Concepts

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

Probability of Success in Binomial Distribution
In the binomial distribution, the "probability of success" is a fundamental aspect. Here, when people talk about success, it means the event you are interested in. To clarify, let's use the given example. Picking a comedy movie from a selection of movies is what we define as a success. The initial probability of selecting a comedy is calculated by dividing the number of comedy movies by the total number of movies.
In this case, it is 5 comedy movies out of 20, so the probability is \( p = \frac{5}{20} = 0.25 \).
  • Success: Picking a comedy movie
  • Failure: Picking a non-comedy movie
  • Initial Probability: 0.25 for picking a comedy
This probability should remain the same for all trials in a true binomial distribution. In our example, this is not upheld because the chances of selecting a comedy change with each selection made.
Independent Trials in Probability
Independent trials mean that the outcome of one trial does not affect the outcome of another. Consider flipping a coin; the result of one flip doesn鈥檛 magically change what happens on the next.

In the context of our movie selection, however, picking one movie changes what's available for the next pick. The pool of options shrinks without replacement, making the trials dependent on each other.

When selecting movies at random, after each pick, movie could be a comedy or not, which changes the total.
  • The initial pool has 20 movies, 5 of which are comedies.
  • Once you pick a movie, you have only 19 left, altering the chance.
  • Binomial distribution requires each trial to be independent, which is not met here.
Fixed Number of Trials Anytime
A fixed number of trials means that the number of experiments or attempts is set beforehand and does not change. In our case, this means selecting four movies from the set of twenty.

For a scenario to fit a binomial distribution pattern, it must repeatedly perform the experiment or trial a certain number of times.
  • In the original setup, we have four distinct selections, which means four trials.
  • Having a clear number of trials helps identify the structure needed for binomial probability calculations.
  • If any of these trials are affected by previous picks, it disrupts the fixed structure.
Although the number of movie selections remains fixed in this situation, the independence criterion is crucially violated, which affects the ability to model the situation using a binomial distribution.

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Most popular questions from this chapter

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In the United States, the mean birth weight for boys is \(3.41 \mathrm{~kg}\), with a standard deviation of \(0.55 \mathrm{~kg}\). (Source: cdc.com.) Assuming that the distribution of birth weight is approximately normal, find the following using a table, calculator, or software. a. A baby is considered of low birth weight if it weighs less than \(2.5 \mathrm{~kg}\). What proportion of baby boys in the United States are born with low birth weight? b. What is the \(z\) -score for a baby boy that weighs \(1.5 \mathrm{~kg}\) (defined as extremely low birth weight)? c. Typically, birth weight is between \(2.5 \mathrm{~kg}\) and \(4.0 \mathrm{~kg}\). Find the probability a baby boy is born with typical birth weight. d. Matteo weighs \(3.6 \mathrm{~kg}\) at birth. He falls at what percentile? e. Max's parents are told that their newborn son falls at the 96 th percentile. How much does Max weigh?
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