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Ask Marilyn: Chances of Drawing a Card
Dan Cowles of Atlanta, Georgia, writes:
Marilyn: A deck of cards is split into two unequal piles and spread face down on a table. Say 39 cards are on one side of the table, and 12 are on the other side. You want to draw the ace of spades. My wife says that the ace is more likely to be in the larger group, and I agree. But she adds that for this reason, drawing from that group gives her a better chance of getting the ace than drawing from the smaller group. That's where I disagree. I say the chances are the same (1 in 52). Yet I sometimes find myself leaning to her way of thinking. I know I'm right, but I'm finding it difficult to demonstrate why. Can you help?
Look at it this way: With 39 cards, the ace is three times as likely to be in the larger group. But you get to draw only one of them, so when the ace is present in the group, your chances are only 1 in 39 of getting it. Yet when the ace is in the smaller group, your chances are 1 in 13 of getting it.
Say you spread out the deck as described and draw a card, repeating the process four times. Lady Luck agrees to participate in the demonstration and slips the ace of spades into the larger group three times and the smaller group once. Each time you draw from the larger group. Your chances of getting the ace are 1/39, 1/39, 1/39 and zero. And if you draw from the smaller group each time, your chances are zero, zero, zero, and 1/13. These are equivalent, so the overall chances are the same.
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