⧼exchistory⧽
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BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

One of the first conditional probability paradoxes was provided by Bertrand.[Notes 1] It is called the Box Paradox. A cabinet has three drawers. In the first drawer there are two gold balls, in the second drawer there are two silver balls, and in the third drawer there is one silver and one gold ball. A drawer is picked at random and a ball chosen at random from the two balls in the drawer. Given that a gold ball was drawn, what is the probability that the drawer with the two gold balls was chosen?

Notes

  1. J. Bertrand, Calcul des Probabilit\'{e}s, Gauthier-Uillars, 1888.
BBy Bot
Jun 09'24

The following problem is called the two aces problem. This problem, dating back to 1936, has been attributed to the English mathematician J. H. C. Whitehead (see Gridgeman[Notes 1]). This problem was also submitted to Marilyn vos Savant by the master of mathematical puzzles Martin Gardner, who remarks that it is one of his favorites.

A bridge hand has been dealt, i.e. thirteen cards are dealt to each player. Given that your partner has at least one ace, what is the probability that he has at least two aces? Given that your partner has the ace of hearts, what is the probability that he has at least two aces? Answer these questions for a version of bridge in which there are eight cards, namely four aces and four kings, and each player is dealt two cards. (The reader may wish to solve the problem with a 52-card deck.)

Notes

  1. N. T. Gridgeman, Letter, American Statistician, 21 (1967), pgs. 38-39.
BBy Bot
Jun 09'24

In the preceding exercise, it is natural to ask “How do we get the information that the given hand has an ace?” Gridgeman considers two different ways that we might get this information. (Again, assume the deck consists of eight cards.)

  • Assume that the person holding the hand is asked to “Name an ace in your hand” and answers “The ace of hearts.” What is the probability that he has a second ace?
  • Suppose the person holding the hand is asked the more direct question “Do you have the ace of hearts?” and the answer is yes. What is the probability that he has a second ace?
BBy Bot
Jun 09'24

Using the notation introduced in Example, show that in the example of Brams and Kilgour, if [math]x[/math] is a positive power of 2, then

[[math]] \frac{p_{x/2}}{p_{x/2} + p_x} = \frac 35\ . [[/math]]

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Using the notation introduced in Example, let

[[math]] p_x = \left \{ \matrix{ \frac 23 \Bigl(\frac 13\Bigr)^k, & \mbox{if}\,\, x = 2^k, \cr 0, & \mbox{otherwise.}\cr }\right. [[/math]]

Show that there is exactly one value of [math]x[/math] such that if your envelope contains [math]x[/math], then you should switch.

BBy Bot
Jun 09'24

(For bridge players only. From Sutherland.[Notes 1]) Suppose that we are the declarer in a hand of bridge, and we have the king, 9, 8, 7, and 2 of a certain suit, while the dummy has the ace, 10, 5, and 4 of the same suit. Suppose that we want to play this suit in such a way as to maximize the probability of having no losers in the suit. We begin by leading the 2 to the ace, and we note that the queen drops on our left. We then lead the 10 from the dummy, and our right-hand opponent plays the six (after playing the three on the first round). Should we finesse or play for the drop?

Notes

  1. E. Sutherland, “Restricted Choice --- Fact or Fiction?”, Canadian Master Point, November 1, 1993.