A set of 13 cards are numbered from 1 to 13. Two cards are drawn successively wi

Question

A set of 13 cards are numbered from 1 to 13. Two cards are drawn successively with replacement. Let E be the event "the second card has a (strictly) larger number than the first card." Find P(E).

Hint: Let i be the number on the first card you draw and Ai be the event "the first card drawn has the number i on it." What is the probability of event Ai? Then what is the probability of drawing a card with a number strictly larger than i on the second draw? For this problem, try counting how many cards have a number larger than i in the deck, and remember that the cards are replaced after the first draw, so the second draw still comes from 13 cards.

Then the probability of drawing card i on the first card, and then drawing a card with a number strictly larger than i on the second draw, is the product of the two probabilities given in the previous paragraph. We will see this in more detail in Chapter 3.

To see how this hint helps, determine what the event EAi (or AiE) describes

Explanation / Answer

P(E) = P(I card less than II card)

Sample space consists of 13(13) = 169

Favourable events are {(1,2) (1,3)...(1,13)

(2,3) (2,4)...(2,13)

...

(12,13)}

Hence n(E) = 12+11+..+1 = 78

Hence prob (E) = 78/169 = 6/13

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