‘Boy or Girl’ Paradox
Problem 1. (10 pts.) ‘Boy or girl’ paradox.
The following pair of questions appeared in a column by Martin Gardner in Scientific
American in 1959.
- Mr. Jones has two children. The older child is a girl. What is the probability that both
children are girls? - Mr. Smith has two children. At least one of them is a boy. What is the probability
that both children are boys?
Be sure to carefully justify your answers.
Problem 2. (10 pts.) The blue taxi.
In a city with one hundred taxis, 1 is blue and 99 are green. A witness observes a hit-andrun by a taxi at night and recalls that the taxi was blue, so the police arrest the blue taxi
driver who was on duty that night. The driver proclaims his innocence and hires you to
defend him in court. You hire a scientist to test the witness’ ability to distinguish blue and
green taxis under conditions similar to the night of accident. The data suggests that the
witness sees blue cars as blue 99% of the time and green cars as blue 2% of the time.
Write a speech for the jury to give them reasonable doubt about your client’s guilt. Your
speech need not be longer than the statement of this question. Keep in mind that most jurors
have not taken this course, so an illustrative table may be easier for them to understand
than fancy formulas.
Problem 3. (10 pts.) Trees of cards.
There are 8 cards in a hat:
{1♥, 1♠, 1♦, 1♣, 2♥, 2♠, 2♦, 2♣}
You draw one card at random. If its rank is 1 you draw one more card; if its rank is two
you draw two more cards. Let X be the sum of the ranks on the 2 or 3 cards drawn. Find
E(X).
Problem 4. (10 pts.) Dice.
There are four dice in a drawer: one tetrahedron (4 sides), one hexahedron (i.e., cube,
6-sides), and two octahedra (8 sides). Your friend secretly grabs one of the four dice at
random. Let S be the number of sides on the chosen die.
What is the pmf of S?
Now your friend rolls the chosen die and without showing it to you rolls it.. Let R be the
result of the roll.
Use Bayes’ rule to find P(S = k | R = 3) for k = 4, 6, 8. Which die is most likely if
R = 3? Terminology: You are computing the pmf of ‘S given R = 3’.
Which die is most likely if R = 6? Hint: You can either repeat the computation in (b),
or you can reason based on your result in (b).
Which die is most likely if R = 7? No computations are needed!
