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Chapter 4: Problem 6

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. Assuming that boys and girls are equally likely, find the probability of a couple having a boy when their third child is born, given that the first two children were both girls.

Short Answer

Expert verified
The probability is 0.5.

Step by step solution

01

Define the problem

Determine the probability of having a boy on the third birth given that the first two children are girls. Since boys and girls are equally likely, each child being a boy or a girl is an independent event with a probability of 0.5.
02

Identify the given information

Given: The first two children are girls. We need to find the probability of the third child being a boy.
03

Understand independent events

The sex of each child is an independent event, meaning the previous births do not affect the outcome of the next birth. Each child is still equally likely to be a boy or a girl.
04

Calculate the probability

Since each childbirth is an independent event with a probability of 0.5, the probability of the third child being a boy is 0.5.

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Key Concepts

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

independent events
When we talk about 'independent events' in probability, it means that the outcome of one event does not affect the outcome of another. For example, if you flip a coin twice, getting heads on the first flip doesn't change the probability of getting heads on the second flip. Each flip is an independent event. In our exercise about a couple having children, the sex of each child is independent of the others. This means that whether the first two children were girls has no impact on the probability of the third child being a boy or a girl. Each childbirth is considered an independent event.
probability of independent events
The 'probability of independent events' concept helps us understand outcomes when events do not influence each other. In such cases, you calculate the probability of each event separately. For a couple having a boy or a girl, since each event (birth of a child) is independent, the probability remains consistent. So, for every child born, there is always a 50% chance the child will be a boy and a 50% chance the child will be a girl. This is why the probability calculation for the sex of the third child is straightforward. Despite the sex of the first two children, the probability that the third child will be a boy remains \[0.5\].
elementary statistics problems
Elementary statistics problems often introduce basic concepts like identification of probability, understanding of independent events, and simple calculations. These problems are designed to build foundational understanding. The exercise provided fits this category because it requires knowledge of basic probability and independent events. By working through such problems, students learn to apply definitions and principles to find solutions easily. This exercise, calculating the probability of a couple having a boy as their third child, is a perfect example of applying these elementary concepts.
probability of a child鈥檚 sex
Understanding the 'probability of a child's sex' incorporates basic biological and statistical knowledge. Human offspring have an equal chance of being either male or female, assuming no external factors affect the outcome. Statistically, this translates to a probability of 0.5, or 50%, for either outcome. This probability remains constant for each birth because each event, i.e., the birth of each child, is independent of the others. In our exercise example, the fact that the first two children are both girls does not change the probability that the third child will be a boy. Therefore, the probability remains \[0.5\], or 50%.

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Most popular questions from this chapter

Find the probability and answer the questions.Men have XY (or YX) chromosomes and women have XX chromosomes. X-linked recessive genetic diseases (such as juvenile retinoschisis) occur when there is a defective \(\mathrm{X}\) chromosome that occurs without a paired \(\mathrm{X}\) chromosome that is not defective. In the following, represent a defective \(X\) chromosome with lowercase \(x,\) so a child with the \(x\) Y or Yx pair of chromosomes will have the disease and a child with \(X X\) or XY or YX or \(x\) X or \(X x\) will not have the disease. Each parent contributes one of the chromosomes to the child.a. If a father has the defective \(x\) chromosome and the mother has good XX chromosomes, what is the probability that a son will inherit the disease? b. If a father has the defective \(x\) chromosome and the mother has good \(X X\) chromosomes, what is the probability that a daughter will inherit the disease? c. If a mother has one defective \(x\) chromosome and one good \(X\) chromosome and the father has good XY chromosomes, what is the probability that a son will inherit the disease? d. If a mother has one defective \(x\) chromosome and one good \(X\) chromosome and the father has good XY chromosomes, what is the probability that a daughter will inherit the disease?

Express the indicated degree of likelihood as a probability value between \(\boldsymbol{0}\) and \(\boldsymbol{I}\).Based on a report in Neurology magazine, \(29.2 \%\) of survey respondents have sleepwalked.

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a. Five 鈥渕athletes鈥� celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can n mathletes be seated at a round table?

Express the indicated degree of likelihood as a probability value between \(\boldsymbol{0}\) and \(\boldsymbol{I}\). Based on an Adecco survey of hiring managers who were asked to identify the biggest mistakes that job candidates make during an interview, there is a \(50-50\) chance that they will identify "inappropriate attire."
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