Eight cards

Probability Level 3

Eight playing cards (2-9 of hearts) are placed in a row face down on the table in a random order:

If the first card is greater than the second card, then what is the probability that the second card is greater than the third card?

Provide your answer to two decimal places.

The answer is 0.33.

1 solution

Geoff Pilling
Aug 1, 2017

All that really matters is the relative order of the first three cards. By symmetry each has a $\frac{1}{3}$ probability of being the smallest of the three. If the first card is greater than the second card, then the probability that the second card is greater than the third card, is just the same as asking the probability that the third card is the smallest of the first three, which is $\boxed{\frac{1}{3}}$

Hmmm I am not really convinced actually. I arrived at 1/6 by writing out the possibilities for the first 3 cards. For example: 9,8, x gives 1/8 1/7 1 and 9,7,x gives 1/8 1/7 5/6 and so on(of course writing it out and summing is a hell of a job, but it van be done quicker)

Peter van der Linden - 3 years, 10 months ago

Ugh I hit post to soon. What I wanted to add is: why does that give a different outcome, or did I make a mistake calculating?

Peter van der Linden - 3 years, 10 months ago

Well, lets see, the possibilities for the first three cards, A, B, and C are:

1) A > B > C
2) A > C > B
3) B > A > C
4) B > C > A
5) C > A > B
6) C > B > A

Six, just like you mention, all equiprobable. However, the only possible ones for which A > B are (1), (2) and (5), three of them, again all equiprobable. And, only one of those has B > C, namely (1). Therefore the probability is one out of three.

Does that make sense?

Geoff Pilling - 3 years, 10 months ago

@Geoff Pilling – Makes sense, but I found the fault in my reasoning: If the first card is greater than the second card. I didn't take that into account in think.

Peter van der Linden - 3 years, 10 months ago

Eight cards

The answer is 0.33.

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