Suppose there are three cards numbered 2, 7, 10, respectively. Suppose you are t

Question

Suppose there are three cards numbered 2, 7, 10, respectively. Suppose you are to be offered these cards in random order. When you are offered a card, you must immediately either accept it or reject it. If you accept a card, the process ends. If you reject a card, then the next card (if there is one) is offered. If you reject the first two cards, you have to accept the final card. You plan to reject the first card offered, and then to accept the next card if and only if its value is greater than the value of the first card. Let X be a number on the card you have accepted in the end. Find the pmf of X and plot the cdf of X.

Explanation / Answer

The first card can be any of 2,7 or 10, with probabilities = 1/3 each.
Now let us see for different cases:

Case: First card = 2
2nd card can be either 7 or 10, in both cases 2nd draw is final.

Hence,
P(Final Card = 7 | first card = 2) = 1/2 ------------------- (1)
P(Final Card = 10 | first card = 2) = 1/2 ------------------- (2)

Case: First card = 7
If 2nd card = 2 then final card = 10
if 2nd card = 10 then also final card is 10 (since 10>7, 2nd card is final)

Hence,
P(Final card = 10| first card = 7) = 1 ------------------------ (3)
P(Final card = 2| first card = 7) = 0 ------------------------ (4)

Case: First Card = 10
If whatever be the 2nd card, 3rd card is final.
Hence, if 2nd card = 2, then final card = 7
and if 2nd card = 7, then final card = 2

Hence,
P(Final card = 7| first card = 10) = 1/2 --------------------- (5)
P(Final card = 2| first card = 10) = 1/2 --------------------- (6)

Since, the total sample space consists of the above 6 cases only, we can conclude that

P(Final Card = 2) = P(Final card = 2| first card = 10)*P(first card = 10) + P(Final card = 2| first card = 7)*P(first card = 7) = 1/2*1/3+0*1/3 = 1/6
P(Final Card = 7) = P(Final card = 7| first card = 10)*P(first card = 10) + P(Final card = 7| first card = 2)*P(first card = 2) = 1/2*1/3+1/2*1/3 = 1/3 = 2/6
P(Final Card = 10) = P(Final card = 10| first card = 7)*P(first card = 7) + P(Final card = 10| first card = 2)*P(first card = 2) = 1*1/3+1/2*1/3 = 1/2 = 3/6

Hence the PMF is as follows:

f(x) = 1/6, x=2
= 1/3, x=7
= 1/2, x=10

And the CDF is as follows:

F(x) = 1/6, x<=2
= 1/2, x<=7 (=1/6+1/3)
=1, x<=10 (= 1/6+1/3+1/2)

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