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The question:

You are playing a card game where the dealer selects three cards from a playing deck; 1 red and two black. The dealer shuffles the three cards and lays them face down next to each other, so that only he knows the position of the red card. He then asks you to guess where the red card is; but he does not turn over your selected card. You win the game if you guess correctly which card is the red card. After your choice of card is made, the dealer turns over one of the cards you did not select, which is a black card (every time the game is played, the card that the dealer will turn over at this stage of the game is a black card). Now the dealer asks if you would like to change your mind. Are you better to change your mind, stick to your first selection, or doesn't it matter? Your answer must be fully justified with mathematical working. Hint: To get a feel for this question, think about the case where there are many cards (say 1000), but still only 1 red card and the dealer overturns all but 2 (998 cards).

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Now I'm confused with how to approach this problem. I have a feeling that you need to gather the probabilities from before the event of the dealer putting the cards face down, and then after choosing a card and having to choose whether or not to change your mind.

Can anyone help me on approaching this question? Would be very appreciative. Thanks.

mr10k
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    This is the same as the Monty Hall problem - look it up (on this site or elsewhere) and you will find heaps of discussion. – David Sep 16 '15 at 06:18

1 Answers1

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There are three cards; 1 red, 2 black, but shuffled and face down.

You either pick a red or you don't; but the result is not revealed.   At least one of the remaining cards is black, and this is turned over.   The last card will either be red or it won't.   What is the probability that it is red?

Warning: The answer is counterintuitive, so think about it.   It may not be what you first suspect.

Hint: Construct a probability tree to help you decide.

Hint 2 Search term: "Monty Hall Problem"

PS: Since you know this is a conditional probability problem, you should know about the law of total probability.   Let $R_p$ be the probability that you pick a red card. Let $R_o$ be the probability that the last card is red.   You want: $$\Bbb P(R_o) = \Bbb P(R_p)\,\Bbb P(R_o\mid R_p)+ \Bbb P(\neg R_p)\,\Bbb P(R_o\mid \neg R_p)$$

Graham Kemp
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  • Cheers for this.

    I had a look at the Monty Hall problem and managed to work out the solution easily enough, although this is a great insight into resolving this question.

     – mr10k Sep 16 '15 at 07:28