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Chapter 17: Problem 13

Although it's not quite true, suppose the probability of having a male child (M) is equal to the probability of having a female child (F). A couple has four children. a. Are they more likely to have FFFF or to have MFFM? Explain your answer. b. Which sequence in part (a) of this exercise would a belief in the law of small numbers cause people to say had higher probability? Explain. c. Is a couple with four children more likely to have four girls or to have two children of each sex? Explain. (Assume the decision to have four children was independent of the sex of the children.)

Short Answer

Expert verified
Both FFFF and MFFM are equally likely. People might believe MFFM is more likely. A couple is more likely to have two boys and two girls.

Step by step solution

01

Understand the Concept

The probability of having a male child is given as 0.5, and the same goes for a female child. Thus, each child is equally likely to be male or female, and these events are independent.
02

Calculate Probability for Individual Sequences

For each sequence of children, the probability is calculated by multiplying the probabilities of each individual child being male (M) or female (F). For FFFF, it is \((0.5)^4 = 0.0625\). For MFFM, it is also \((0.5)^4 = 0.0625\).
03

Compare Probabilities for FFFF and MFFM

Since both sequences, FFFF and MFFM, have the same probability of occurring, neither is more likely than the other.
04

Discuss the Law of Small Numbers

The law of small numbers refers to the tendency to expect "average" outcomes over small sample sizes. People might think MFFM is more likely because it seems more balanced.
05

Probability of Having Four Girls

The probability of having four girls is the same as the probability of sequence FFFF, \((0.5)^4 = 0.0625\).
06

Probability of Having Two Boys and Two Girls

We must count the different sequences this can occur: MMFF, MFMF, MFFM, FMFM, FMMF, FFMM. There are 6 such sequences. Thus, the probability is \( 6 \times (0.5)^4 = 0.375\).
07

Compare Probabilities in Part c

The probability of having four girls is 0.0625, while the probability of having two boys and two girls is 0.375. Thus, a couple is more likely to have two children of each sex.

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Key Concepts

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

Law of Small Numbers
The Law of Small Numbers is a common cognitive bias. It is the mistaken belief that small samples will reflect the expected average values. This means people may inaccurately assume that short sequences of events, like coin tosses or child births, will show the same balanced characteristics as larger samples.
For instance, in a family planning scenario, someone influenced by this belief might think that sequences like two boys and two girls occur more often simply because it seems 'balanced.' However, it's crucial to remember that in probability, small samples can vary greatly and are not always reflective of their theoretical averages.
  • Common Misconception: Thinking small samples should resemble the population average.
  • Real-World Example: Assuming a couple with four kids will certainly have two of each gender.
  • Important Note: Outcomes like FFFF or MFFM are equally probable regardless of how 'unbalanced' they appear.
Independent Events
In probability theory, two events are independent if the outcome of one does not influence the outcome of the other. When considering the sex of children, the decision for one child does not affect the others. Hence, each child's probability remains constant at 0.5 for being male or 0.5 for being female.
This independence means that for each child born, there's an equal and independent chance of being either female or male, unaffected by preceding outcomes.
  • Key Idea: Independent events do not impact each other.
  • Real-Life Example: Each child's gender is independent of their siblings.
  • Probability Calculation: For four children, calculate as \(0.5^4\) because each birth is an independent event.
Binomial Distribution
The Binomial Distribution is a mathematical model that describes the number of successes in a sequence of independent experiments. Each experiment can result in a success or a failure. Here, defining success as having a female child, each child has a binary outcome (F for female or M for male).
In our example, choosing four children as trials, with an equal chance of male or female, follows a Binomial Distribution.
  • Key Characteristics: Fixed number of trials (four children), two possible outcomes per trial (male or female), and constant probabilities throughout.
  • Probability of Specific Events: All girls (FFFF) is \( (0.5)^4 = 0.0625 \), while two boys and two girls has multiple sequences, amounting to \( 6 imes (0.5)^4 = 0.375 \).
  • Expectation vs. Reality: Larger sample sizes tend toward the expected ratio, but smaller ones can vary more widely.

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Most popular questions from this chapter

Many people claim that they can often predict who is on the other end of the phone when it rings (obviously without caller ID). Do you think that phenomenon has a normal explanation? Explain.

You are making a hotel reservation and are offered a choice of two rates. The advanced purchase rate is \(\$ 100,\) but your credit card will be charged immediately and there is no refund, even if you don't use the room. The flexible rate is \(\$ 140\) but you don't pay anything if you don't use the room. Suppose \(p\) is the probability that you will end up using the room. a. Suppose \(p=0.70,\) so there is a \(70 \%\) chance you will use the room. What is the expected value of your cost if you reserve the room with the flexible rate? (Hint: What are the two possible amounts you could pay, and what are their probabilities?) b. No longer assume a specific value for \(p .\) In terms of \(p,\) what is the expected value of your cost if you reserve the room with the flexible rate? c. What is the expected value of your cost if you choose the advanced purchase rate? (Hint: There is only one possible amount.) d. For what value of \(p\) are the expected values you found in parts (b) and (c) the same? e. For what range of values of \(p\) are you better off choosing the advanced purchase rate?

Explain why it would be much more surprising if someone were to flip a coin and get six heads in a row after telling you they were going to do so than it would be to simply watch them flip the coin six times and observe six heads in a row.

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A statistics professor once made a big blunder by announcing to his class of about 50 students that he was fairly certain that someone in the room would share his birthday. We have already learned that there is a \(97 \%\) chance that there will be two people in a room of 50 with a common birthday. Given that information, why was the professor's announcement a blunder? Do you think he was successful in finding a match? Explain.
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