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Chapter 8: Problem 7

Assuming that people are equally likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what鈥檚 the probability of (a) the husband being born during January and the wife being born during February? (b) both husband and wife being born during December? (c) both husband and wife being born during the spring (April or May)? (Hint: First, find the probability of just one person being born during April or May.)

Short Answer

Expert verified
The probabilities of (a), (b), and (c) are \( \frac{1}{144} \), \( \frac{1}{144} \), and \( \frac{1}{36} \) respectively.

Step by step solution

01

Calculate the probability for (a)

For (a), the goal is to find the probability of the husband being born in January and the wife in February. Since these are independent events, the probability is the product of the individual probabilities. Thus, calculate the product of the probability of being born in January (1/12) and the probability of being born in February (1/12). This gives \( \left( \frac{1}{12} \right) ^2 = \frac{1}{144} \).
02

Calculate the probability for (b)

For (b), the goal is to find the probability of both the husband and wife being born in December. Again, since these are independent events, the probability is the product of the individual probabilities. Thus, calculate the product of the probability of being born in December for both husband and wife. This gives \( \left(\frac{1}{12} \right)^2 = \frac{1}{144} \).
03

Calculate the probability for (c)

For (c), the goal is to find the probability of both the husband and wife being born in the spring - i.e. either April or May. First, find the probability of one person being born in April or May. Since these months are equally likely and independent, the probability is the sum of the individual probabilities. This gives \( \frac{1}{12} + \frac{1}{12} = \frac{1}{6} \). To find the probability of both being born in these months, calculate the square of this probability, which gives \( \left( \frac{1}{6} \right)^2 = \frac{1}{36} \).

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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, when we talk about independent events, we mean that the occurrence of one event does not affect the probability of the other event happening. For example, in the context of the exercise, the birth month of a husband doesn't influence the birth month of his wife.
This assumption of independence allows us to calculate the combined probability of two independent events occurring together by simply multiplying the individual probabilities of each event.
  • To find the probability of the husband being born in January and the wife being in February, we multiply the probability of each independent event: \( rac{1}{12} \times rac{1}{12} = rac{1}{144} \).
  • Similarly, for both being born in December, it's again \( rac{1}{12} \times rac{1}{12} = rac{1}{144} \).
Understanding that these events are independent makes calculations straightforward, as you're simply multiplying the likelihood of two separate events.
Birth Month Probability
Birth month probability refers to the chance of a person being born in a specific month. Assuming every month is equally likely to host a birth, the probability of any individual being born in a specific month, like January or February, is \( \frac{1}{12} \).
This simple fraction arises from the 12 months in a year.
When events are independent, as in the birth months of a couple, the probabilities remain consistent.
  • For any given month, the probability of one person's birth is always \( \frac{1}{12} \).
  • The exercise leverages this uniform distribution, whether it's calculating the chances for January, February, or even December.
This general approach simplifies probability problems that involve specific birth months.
Spring Birth Probability
Spring birth probability is about understanding the likelihood of being born in months traditionally considered part of spring, like April and May. To determine this, we first calculate the individual probability for each of these months.
Since each month's probability is \( \frac{1}{12} \), the combined probability for being born in either April or May is \( \frac{1}{12} + \frac{1}{12} = \frac{1}{6} \).But what if we want to find out the likelihood of both spouses being born in spring?
Given these are independent events, we square the combined probability: \( \left( \frac{1}{6} \right)^2 = \frac{1}{36} \).
  • The probability of one person being born in spring is calcualted by summing the probabilities of birth in April and May.
  • The joint probability for two independent spring births multiplies the individual probability by itself.
This approach helps you calculate probabilities when dealing with specific seasons or clusters of months.

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Most popular questions from this chapter

The probability of a boy being born equals \(.50,\) or \(1 / 2\), as does the probability of a girl being born. For a randomly selected family with two children, what's the probability of (a) two boys, that is, a boy and a boy? (Reminder: Before using either the addition or multiplication rule, satisfy yourself that the various events are either mutually exclusive or independent, respectively.) (b) two girls? (c) either two boys or two girls?

Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during (a) June? (b) any month other than June? (c) either May or June?

Television stations sometimes solicit feedback volunteered by viewers about a televised event. Following a televised debate between Barack Obama and Mitt Romney in the 2012 presidential election campaign, a TV station conducted a telephone poll to determine the "winner." Callers were given two phone numbers, one for Obama and the other for Romney, to register their opinions automatically. (a) Comment on whether or not this was a random sample. (b) How might this poll have been improved?

Continue to assume that people are equally likely to be born during any of the months. However, just for the sake of this exercise, assume that there is a tendency for married couples to have been born during the same month. Furthermore, we wish to calculate the probability of a husband and wife both being born during December. (a) It would be appropriate to use the multiplication rule for independent outcomes. True or False? (b) The probability of a married couple both being born during December is smaller than, equal to, or larger than \((1 / 12)(1 / 12)=1 / 144 .\) (c) With only the previous information, it would be possible to calculate the actual probability of a married couple both being born during December. True or False?

As subjects arrive to participate in an experiment, tables of random numbers are used to make random assignments to either group A or group B. (To ensure equal numbers of subjects in the two groups, alternate subjects are automatically assigned to the other, smaller group.) Indicate with a Yes or No whether each of the following rules would work: (a) Assign the subject to group \(\mathrm{A}\) if the random number is even and to group \(\mathrm{B}\) if the random number is odd. (b) Assign the subject to group A if the first digit of the random number is between \(\mathbf{O}\) and 4 and to group \(\mathrm{B}\) if the first digit is between 5 and \(9 .\) (c) Assign the subject to group A if the first two digits of the random number are between 00 and 40 and to group \(\mathrm{B}\) if the first two digits are between 41 and \(99 .\) (d) Assign the subject to group A if the first three digits of the random number are between 000 and 499 and to group \(\mathrm{B}\) if the first three digits are between 500 and \(999 .\)
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