A gambler has a proposition for you.
In this scenario, let’s say you are always willing to take a bet on 50-50 odds or better, but you will never take a bet when the odds are against you.
First the gambler separates the four aces, shuffles them, and deals you one card from the stack of four. “Would you be willing to bet that you have a diamond?” he asks. “Of course not,” you say. “There is only a 1 in 4 chance, or 25 percent.” The gambler then deals you a second card from the three remaining aces. “How about with your odds doubled?” he asks. “OK,” you say. “Now the odds are 50 percent, and I will take a bet with 50-50 odds.”
The gambler then replaces the aces in the deck, shuffles the full deck of 52 cards, and then deals you a single card. “I assume you still will not bet that you have a diamond,” he says. “That’s right,” you say. “It’s still a 25 percent chance.” The gambler then deals you a second card. “How about with your odds doubled? You’ll take this bet, correct?”
Is the gambler right about your chances? Should you take the bet?
See AnswerA simple way to determine that the gambler is not speaking the truth is to continue comparing the four aces to the full deck. With only the four aces, as you are dealt more cards, the odds that you have a diamond are easy to calculate: with 1 card, you have a 25 percent chance; with two cards, 50 percent, three cards, 75 percent; four cards, 100 percent. On the other hand, when you are dealt three cards from a full deck, it does not seem right that you could have a 75 percent chance of having a diamond, and clearly when you are dealt four cards from a full deck, the odds cannot be 100 percent that you have a diamond.
So, when you are dealt two cards from a deck of 52, what are the odds that you have a diamond? To calculate the odds properly, we need to consider two different probabilities: that the first card is a diamond (which is correctly stated as 13/52 or 1/4) and that the second card is a diamond when the first card has not been replaced. This means that for the second card, our probability will not be out of 52 cards but out of 51.
One elegant solution is to think instead of the probability of not drawing a diamond, and then subtract that from 1. So to calculate the odds that neither card is diamond, first take the odds of not drawing a diamond with the first card (39/52 or 3/4) and multiply it by the odds of not drawing a diamond from the 38 remaining non-diamonds in the deck of 51 (38/51). Multiplying 3/4 by 38/51 gives us 114/204, which simplifies to 19/34. Subtract that from 1 and we have our odds: 15/34 or about 44 percent.
When you are dealt three cards from a deck of 52, the odds you have a diamond are about 59 percent; with four cards, the odds are about 70 percent.