(Corrections invited.)
This week’s "Ask Marilyn" column has a problem for which Marilyn gives the correct answer for the wrong reason. The question is, "Is being dealt two aces or an ace and a deuce more probable." There are a couple of unstated assumptions: using ordinary deck and dealing exactly two cards are the main ones. Marilyn’s explanation is to use the tableau:
AA
22
then to point out that being dealt two aces requires being dealt the top row and being dealt ace/deuce is being dealt one of the columns. This explanation is wrong on two points. First, it implies that the odds for the A2 combination are twice that of the AA combination (actually, the odds are 8:3 in favor A2 over AA) and it uses an arrangement of the ex post dealt cards rather than ex ante undealt deck.
The correct explanation: for two aces, the probability of an ace on the first card is 4/52 and the probability of an ace on the second card is 3/51; as these both must occur, the probability of both occurences is 12/2652. For an ace then a deuce, one has a probability of 4/52 for the ace and 4/51 for the deuce giving 16/2652 plus the (identical) probabilities of a deuce then an ace giving 32/2652 for the probability of the A2 pair.
Of course, naming the cards makes the probabilities equal. That is, the ace of spades and deuce of hearts has the same probability as the ace of hearts and the ace of diamonds.
Statistics: Posted by Doctor Stochastic — Sat Jan 09, 2010 6:56 pm