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Chapter 14: Problem 13

Larry reads that half of all super jumbo eggs contain double yolks. So he always buys super jumbo eggs and uses two whenever he cooks. If eggs do or don't contain two yolks independently of each other, the number of eggs with double yolks when Larry uses two chosen at random has the distribution a. binomial with \(n=2\) and \(p=1 / 2\). b. binomial with \(n=2\) and \(p=1 / 3\). c. binomial with \(n=3\) and \(p=1 / 2\).

Short Answer

Expert verified
The correct answer is (a): binomial with \(n=2\) and \(p=1/2\).

Step by step solution

01

Identify the Scenario

Larry buys super jumbo eggs, where each egg has a probability of 0.5 of containing a double yolk. He typically uses two eggs at a time. We need to determine the distribution of the number of double yolk eggs when two are chosen at random.
02

Recognize the Distribution

Since each egg has a probability of being a double yolk and the outcomes are independent, this problem is a perfect fit for a binomial distribution, where the number of trials is the number of eggs used.
03

Confirm Number of Trials

Larry always uses two eggs, which means the number of trials, denoted by \(n\), is 2.
04

Determine Probability of Success

The probability of an egg having a double yolk (success) is given as \(p = \frac{1}{2}\).
05

Identify the Binomial Distribution

With \(n=2\) and \(p=\frac{1}{2}\), the number of eggs with double yolks when Larry uses two eggs is binomially distributed. Thus, the correct answer matches option (a), which states it is a binomial distribution with \(n=2\) and \(p=\frac{1}{2}\).

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Key Concepts

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

Probability
Probability helps us quantify uncertainty. It's a way to express how likely an event is to happen. In Larry's case, the probability of each super jumbo egg having a double yolk is given as 0.5, or 50%.

This means, out of 100 such eggs, roughly 50 will contain double yolks. When dealing with probability:
  • The sum of probabilities of all possible outcomes is always 1.
  • Probability is represented on a scale from 0 to 1, where 0 means the event will not happen, and 1 means it will definitely happen.

In calculations, finding probability often involves determining the number of favorable outcomes divided by the total possible outcomes.

In Larry's case, since each egg is independent, this probability applies to both eggs he uses in each cooking session.
Independent Events
Independent events occur when the outcome of one event does not affect the outcome of another. In Larry's scenario, each super jumbo egg's chance of having a double yolk is independent of the other.

This means, the probability of the first egg having a double yolk doesn't change the probability of the second egg being a double yolk. Independence is a key concept in probability and statistical experiments. Some features include:
  • If events A and B are independent, then the probability of A and B occurring together, denoted as P(A and B), is the product of their probabilities: \(P(A) \times P(B)\).
  • Independent events don't provide information about each other - knowing the outcome of one doesn't help us predict the other.

For Larry, each time he uses two eggs, whether one has a double yolk does not affect the other. This forms the basis of his binomial distribution problem.
Random Variables
Random variables are a fundamental concept in probability and statistics. A random variable is essentially a numerical description of the outcome of a random event.

In the problem, the random variable can be defined as the number of double yolk eggs when Larry uses two eggs. It can take values such as 0, 1, or 2 depending on how many eggs turn out to have double yolks.

Important points about random variables include:
  • Random variables can be discrete or continuous. In Larry's case, the random variable is discrete as it takes on whole number values.
  • They are often denoted by capital letters such as X, Y, etc., where each value corresponds to an event's outcome.

Understanding how random variables work simplifies calculating probabilities related to them, especially in distributions like the binomial distribution.
Statistics Education
Statistics education involves teaching the methods for collecting, analyzing, interpreting, and presenting data. It helps students make informed decisions based on data.

Learning about binomial distribution, as in Larry's example, is an essential part of statistics education. This type of distribution models numerous real-world situations where outcomes are binary (success/failure).

Key aspects include:
  • Understanding distributions helps predict frequencies of outcomes over time.
  • Statistical literacy empowers individuals to critically analyze reports and statistics they encounter in everyday life.

Through exercises like Larry鈥檚 problem, students practice recognizing patterns and applying statistical methods, constructing a foundation for more complex topics encountered in real-world data analysis.

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Most popular questions from this chapter

Roulette-Betting on Red. A roulette wheel has 38 slots, numbered 0,00 , and 1 to 36 . The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. a. If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? b. You decide to play roulette four times, each time betting on red. What is the distribution of \(X\), the number of times you win? c. If you bet the same amount on each play and win on exactly two of the four plays, then you will "break even." What is the probability that you will break even? d. If you win on fewer than two of the four plays, then you will lose money. What is the probability that you will lose money?

Checking for Survey Errors. One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population. About \(24 \%\) of the Canadian population is first generation-that is, they were born outside Canada. \(\underline{6}\) The number \(X\) of first-generation Canadians in random samples of 1500 persons should therefore vary with the binomial ( \(n=1500, p=0.24\) ) distribution. a. What are the mean and standard deviation of \(X\) ? b. Use the Normal approximation to find the probability that a sample will contain between 340 and 390 firstgeneration Canadians. Check that you can safely use the approximation.

In a group of 10 college students, three are psychology majors. You choose three of the 10 students at random and ask their major. The distribution of the number of psychology majors you choose is a. binomial with \(n=10\) and \(p=0.3\). b. binomial with \(n=3\) and \(p=0.3\). c. not binomial. In a test for ESP (extrasensory perception), a subject is told that cards that the experimenter can see, but that the subject cannot see, contain either a star, a circle, a wave, or a square. As the experimenter looks at each of 4 cards in turn, the subject names the shape on the card. A subject who is just guessing has probability \(0.25\) of guessing correctly on each card. Questions \(14.16\) to \(\underline{14.18}\) use this information.

Response Rates for Random Digit Dialing. When an opinion poll uses random digit dialing to select respondents for polls, the response rate is approximately \(10 \%\) for people contacted by cell phone. You watch a pollster dial 20 cell numbers using random digit dialing. a. What is the probability that exactly two calls yield a response? b. What is the probability that at most two calls yield a response? c. What is the probability that at least two calls yield a response? d. What is the probability that fewer than two calls yields a response? e. What is the probability that more than two calls yield a response?

Response Rates for Random Digit Dialing. When an opinion poll uses random digit dialing to select respondents for polls, the response rate (the percentage who actually provide a usable response to the poll) is approximately \(10 \%\) for people contacted by cell phone. \(-\) A pollster dials 20 cell phone numbers. \(X\) is the number that respond to the pollster.
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