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Chapter 6: Problem 59

A married couple plans to have four children, and they are wondering how many boy they should expect to have. Assume none of the children will be twins or other multiple births. Also assume the probability that a child will be a boy is \(0.50\). Explain why this is a binomial experiment. Check all four required conditions.

Short Answer

Expert verified
The expected number of boys in the family given these conditions is 2.

Step by step solution

01

Fixed Number of Trials

In this case, the fixed number of trials is the four children the couple plans to have. This means that the experiment will be repeated a fixed number of times.
02

Binary Outcomes

Each child born can either be a boy or not a boy (i.e., a girl). These are the only two possibilities and these outcomes are mutually exclusive, meaning one excludes the possibility of the other.
03

Independence

The sex of each child is independent of the sex of the other children. The gender of a future child is not influenced by the genders of previous children.
04

Fixed Probability of Success

The probability of having a boy, labelled as success in this context, is given as 0.50. This probability is the same for each trial (child). This complies with the requirement of a fixed probability of success.
05

Calculating Expectation

The expected value or mean of a binomial distribution is calculated using the formula \(E(x) = np\), where \(n\) is the number of trials and \(p\) is the probability of success. Here, \(E(x) = 4 * 0.50 = 2\).

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Key Concepts

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

Fixed Number of Trials
In a binomial experiment, having a fixed number of trials is essential. This simply means we plan to repeat the same experiment a certain number of times. Here, the couple plans to have four children, which can be considered as four separate trials.
Each trial corresponds to the birth of a child.
  • The number of trials is predetermined. In this example, it is exactly four.
  • No additional trials will be added after the four children are planned and born.
By clearly setting the number of trials to four, the scenarios this couple is exploring matches the first condition needed for a binomial experiment.
Binary Outcomes
A key characteristic of a binomial experiment is that each trial should result in just one of two possible outcomes. These outcomes need to be mutually exclusive, meaning if one happens, the other cannot. Here, each child born can either be a boy or a girl.
  • These are the only possible outcomes, covering all possibilities.
  • They do not overlap; a child cannot be both a boy and a girl.
In each trial of having a child, the only relevant outcomes are boy or not a boy (girl). This fulfills the second condition for it to be called a binomial experiment.
Independent Trials
For a series of trials to be used in a binomial experiment, each trial must be independent of the others. In this example, this requirement implies that each child's gender does not affect the gender of any other children.
  • The outcome of one trial (having one child) does not impact the outcomes of any other trials.
  • There isn't any influence between the gender of one child and any siblings born after.
The independence of each trial ensures randomness in the experiment, a quintessential feature of binomial experiments.
Fixed Probability of Success
The fixed probability of success criterion in a binomial experiment dictates that each trial should have the same likelihood of resulting in a 'success'. In this scenario, success is defined as having a boy. The probability of having a boy is given as 0.50, which remains constant for each child.
  • The probability is not affected by previous trials.
  • Each trial (birth) consistently retains the 0.50 probability of success.
With the probability staying fixed across all trials, this example satisfies the final condition necessary for a binomial experiment.

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Most popular questions from this chapter

According to dogtime .com, the mean weight of an adult St. Bernard dog is 150 pounds. Assume the distribution of weights is Normal with a standard deviation of 10 pounds. a. Find the standard score associated with a weight of 170 pounds. b. Using the Empirical Rule and your answer to part a, what is the probability that a randomly selected St. Bernard weighs more than 170 pounds? c. Use technology to confirm your answer to part \(\mathrm{b}\) is correct. d. Almost all adult St. Bernard's will have weights between what two values?

The length of gestation for hippopotami is approximately Normal, with a mean of 270 days and a standard deviation of 7 days. a. What percentage of hippos have a gestation period less than 260 days? b. Complete this sentence: Only \(6 \%\) of hippos will have a gestational period longer than ____days. c. In 2017 , Fiona the Hippo was born at the Cincinnati Zoo, 6 weeks premature. This means her gestational period was only about 228 days. What percentage of hippos have a gestational period of 228 days or less?

Alaska Airlines has an on-time arrival rate of \(88 \%\). Assume that in one day, this airline has 1200 flights. Suppose we pick one day in December and find the number of ontime Alaska Airline arrivals. Why would it be inappropriate to use the binomial model to find the probability that at least 1100 of the 1200 flights arrive on time? What condition or conditions for use of the binomial model is or are not met?

For each situation, identify the sample size \(n\), the probability of a success \(p\), and the number of success \(x\). When asked for the probability, state the answer in the form \(b(n, p, x)\). There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. Since the Surgeon General's Report on Smoking and Health in 1964 linked smoking to adverse health effects, the rate of smoking the United States have been falling. According to the Centers for Disease Control and Prevention in 2016, \(15 \%\) of U.S. adults smoked cigarettes (down from \(42 \%\) in the \(1960 \mathrm{~s}\) ). a. If 30 Americans are randomly selected, what is the probability that exactly 10 are smokers? b. If 30 Americans are randomly selected, what is the probability that exactly 25 are not smokers?

According to a survey conducted by OnePoll, a marketing research company, \(10 \%\) of Americans have never traveled outside their home state. Assume this percentage is accurate. Suppose a random sample of 80 Americans is taken. a. Find the probability that more than 12 have never travelled outside their home state. b. Find the probability that at least 12 have never travelled outside their home state. c. Find the probability that at most 12 have never travelled outside their home state.
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