Seven good shuffles does a very good job of randomizing a deck of cards. That is

Question

Seven good shuffles does a very good job of randomizing a deck of cards. That is, no matter the initial order of the stack, after seven shuffles, the permutations of the cards ( top card, second card, etc) are roughly equally likely

a, Suppose that 3 times in a row you randomize a deck of 52 cards, observe the top card and return it to the deck. What is probability that all three of your observation match?

b, Suppose that you draw 4 cards from a randomized deck ( any set of 4 is equally likely). What is the probability all 4 are all from the same suit?

c, Suppose hat you draw 4 cards from a randomized deck ( any set of 4 is equally likely). What is the probability all 4 are all from different suits?

Explanation / Answer

A)

The probability that a card is the top card is 1/52.

Hence, it occuring three times is (1/52)^3.

However, we have 52 possible cards to do it, so the probability is

P = 52*(1/52)^3 = 0.000369822 [ANSWER]

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b)

There are 52C4 = 270725 ways to choose any 4 cards.

There are 13C4 = 715 ways to choose 4 cards of a suit. As there are 4 suits, 4*715 = 2860 ways to do the job.

hence,

P = 2860/270725 = 0.010564226 [ANSWER]

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c)

There are 13*13*13*13 = 28561 ways to do the job.

Hence,

P = 28561/270725 = 0.105498199 [ANSWER]

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