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Chapter 4: Problem 7

In a 1974 "Dear Abby" letter, a woman lamented that she had just given birth to her eighth child and all were girls! Her doctor had assured her that the chance of the eighth child being a girl was less than 1 in 100 . What was the real probability that the eighth child would be a girl? (A) 0.0039 (B) 0.5 (C) \((0.5)^{7}\) (D) \((0.5)^{8}\) (E) \(\frac{(0.5)^{7}+(0.5)^{8}}{2}\)

Short Answer

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B

Step by step solution

01

Understand the problem

Identify what is being asked: We need to determine the probability that the eighth child is a girl. Ignore the information about the outcomes of the previous seven children.
02

Identify relevant probabilities

Each child's gender is an independent event with two possible outcomes: girl or boy. The probability of being a girl (or boy) for each child is therefore 0.5 (or 50%).
03

Calculate the probability for the eighth child

The probability that the eighth child is a girl is simply the probability of one independent event occurring. This probability is 0.5 or 50%, regardless of the genders of the previous seven children.
04

Match with the given options

Among the provided options, the probability that the eighth child is a girl matches with option (B) 0.5.

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Key Concepts

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

independent events
In probability theory, independent events are events where the occurrence of one event does not affect the probability of the other. This concept is crucial in many real-world scenarios, including genetics and gender probability. For instance, when considering the likelihood of a child being a girl or a boy, each child's gender is typically an independent event.
This means that the gender of one child does not influence the gender of subsequent children. Therefore, the probability of each child being a girl remains constant at 0.5, regardless of how many girls or boys have already been born.
probability theory
Probability theory is a branch of mathematics concerned with analyzing random phenomena. It provides the tools needed to assess and predict various outcomes. Probability values range from 0 to 1, where 0 indicates impossibility, and 1 signifies certainty.
Evaluating the probability of a single event within a given framework is a fundamental aspect of probability theory. For instance, with gender probability, because the events are independent, the probability of having a girl (P(Girl)) remains constant at 0.5 for each child. Detailed knowledge of this theory can assist in making informed predictions and decisions in diverse areas such as medicine, economics, and everyday life.
statistical outcomes
Statistical outcomes refer to the possible results that can occur when an experiment or trial is conducted. Each specific outcome is a realization of a random event. For example, when a mother gives birth, the result is either a boy or a girl, each with a certain probability.
In the context of the original exercise, the key is to understand that each birth is an independent event. This means the outcome (boy or girl) for each birth happens independently of previous births, with a constant probability of 0.5 for each outcome.
gender probability
Gender probability in the context of childbirth refers to the chance that a child will be either a girl or a boy. Typically, and assuming equal biological conditions, this probability is considered to be 0.5 (or 50%) for a girl and 0.5 (or 50%) for a boy.
It's crucial to remember that each birth is an independent event. Thus, even if a family has previously had seven girls, the probability of having an eighth girl remains at 0.5. This principle of independent events ensures that the gender of each child is not influenced by the genders of previous children.

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Most popular questions from this chapter

Can the function \(f(x)=\frac{x+6}{24},\) for \(x=1,2,\) and \(3,\) be the probability distribution for some random variable taking the values \(1,2,\) and \(3 ?\) (A) Yes. (B) No, because probabilities cannot be negative. (C) No, because probabilities cannot be greater than 1 . (D) No, because the probabilities do not sum to 1. (E) Not enough information is given to answer the question.

Mathematically speaking, casinos and life insurance companies make a profit because of (A) their understanding of sampling error and sources of bias. (B) their use of well-designed, well-conducted surveys and experiments. (C) their use of simulation of probability distributions. (D) the central limit theorem. (E) the law of large numbers.

Consider the following table of ages of U.S. senators: $$\begin{array}{|l|l|l|l|l|l|l|}\hline \text { Age (yr): } & <40 & 40-49 & 50-59 & 60-69 & 70-79 & >79 \\\\\hline \text { Number of senators: } & 5 & 30 & 36 & 22 & 5 & 2\\\\\hline\end{array}$$ What is the probability that a senator is under 70 years old given that he or she is at least 50 years old? (A) 0.580 (B) 0.624 (C) 0.643 (D) 0.892 (E) 0.969

Suppose that, for any given year, the probabilities that the stock market declines, that women's hemlines are lower, and that both events occur are, respectively, \(0.4,0.35,\) and \(0.3 .\) Are the two events independent? (A) Yes, because \((0.4)(0.35) \neq 0.3 .\) (B) No, because \((0.4)(0.35) \neq 0.3\). (C) Yes, because \(0.4>0.35>0.3\). (D) No, because \(0.5(0.3+0.4)=0.35\). (E) There is insufficient information to answer this question.

Suppose that one out of 20 apples from a particular orchard is wormy. What are the mean and standard deviation for the number of apples to be sampled from this orchard before finding a wormy apple? (A) Mean \(=5,\) standard deviation \(=1\) (B) Mean \(=5,\) standard deviation \(=1-0.05\) (C) Mean \(=10,\) standard deviation \(=\sqrt{\frac{(0.05)^{2}}{1-0.05}}\) (D) Mean \(=20,\) standard deviation \(=\sqrt{380}\) (E) Mean \(=20,\) standard deviation \(=\sqrt{\frac{(0.05)^{2}}{1-0.05}}\)
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