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Contestants in a game show are asked to from a line. One by one, each of them will be given one card (face up) from a deck of 52 until someone gets either an Ace of Spades or an Ace of Clubs. The first person who gets one of those cards will receive a sum of 1 million dollars.

Before the game begins, you have the chance of choosing where in the line to position yourself in. How will you find out where to which position will be the one most likely to be the first to receive one of the 2 black Ace cards.

My first thought was to do a mock version of this and count how many cards were laid out before the first black Ace appeared. I repeated this for 25 times while recording the results for each trial to hopefully get an estimate to the average number of cards distributed before one of the black Aces appeared.

Parcly Taxel
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James Calase
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    What have you tried? – Michael Burr Nov 28 '20 at 05:58
  • My first thought was to do a mock version of this and count how many cards were laid out before the first black Ace appeared. I repeated this for 25 times while recording the results for each trial to hopefully get an estimate to the average number of cards distributed before one of the black Aces appeared. – James Calase Nov 28 '20 at 06:05
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    I like the experimental approach, but I suspect you would need to do thousands of trials in this case to ascertain the correct answer with confidence... some of the probabilities are quite similar. – mjqxxxx Nov 28 '20 at 06:28

2 Answers2

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  • The first-in-line is the winner with probability $\frac2{52}$
  • The second-in-line is the winner with probability $\frac{50}{52}×\frac2{51}$
  • In general, the $n$th-in-line is the winner with probability $$f(n)=\frac2{53-n}\prod_{k=1}^{n-1}\frac{51-k}{53-k}$$

However, $f(n)$ is strictly decreasing because (as can be shown by manipulating the definition) $$f(n+1)=\frac{51-n}{52-n}f(n)$$ Hence you should ask to be first-in-line to have the highest probability of winning.

Parcly Taxel
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Think of it this way. If there were only one winning card, it would have equal probability to be in each place in the deck, and (equivalently) each person in line would be equally likely to receive the winning card. With multiple winning cards, it’s still the case that each person in line is equally likely to receive a winning card; but to win, you need to receive a winning card and be the first to do so. This is most likely if you’re first in line, since the second condition is then guaranteed.

mjqxxxx
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