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Chapter 5: Problem 29

Which of the following are binomial experiments? Explain why. a. Drawing 3 balls with replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls b. Drawing 3 balls without replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls c. Selecting a few households from New York City and observing whether or not they own stocks when it is known that \(28 \%\) of all households in New York City own stocks

Short Answer

Expert verified
Only the experiment described in point A fits all the criteria for a binomial experiment, while the procedures described in point B and C do not.

Step by step solution

01

Analyze Case A

The experiment consists of drawing 3 balls with replacement from a box that contains 10 balls, 6 of which are red and 4 are blue. Each draw results in two possible outcomes (red or blue), the trials (drawing the balls) are independent because after each draw the ball is replaced back, and the probability for each outcome is constant (60% chance of getting red and 40% chance of getting blue). Therefore, this is a binomial experiment.
02

Analyze Case B

In this experiment, the procedure is as in case A, but the balls are drawn without replacement. The fact that we are drawing without replacement makes the trials dependent (because the outcome of the first draw influences the probability of the outcomes in the following draws). Therefore, this is not a binomial experiment.
03

Analyze Case C

When selecting a few households from New York City and observing whether or not they own stocks, there are two possible outcomes for each trial (the household owns stocks or it doesn鈥檛) and each trial is independent. However, it is difficult to guarantee that the probability of 'success' (a household owning stocks) would remain constant at 28% unless the sample size is a large portion of the population, which gets unpractical. Therefore, this is not likely a binomial experiment.

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