(Combinations) Consider the following problem: \"From an ordinary deck of 52 car

Question

(Combinations) Consider the following problem: "From an ordinary deck of 52 cards, seven cards are drawn at random and without replacement. What is the probability that at least one of the cards is a King?" A student in solves this problem as follows: To make sure there is a least one King among the seven cards drawn, first choose a King; there are C(4,1) possibilities; then choose the other six cards from the 51 cards remaining in the deck, for which there are C(51,6) possibilities. Thus, the solution is C(4,1)*C(51,6) / C(52,7) = 0.5385.

However, upon testing the problem experimentally, the student finds that the correct answer is somewhat less, around 0.45.

(a) Calculate the correct answer using combinations

(b) Explain carefully why the student's solution is incorrect.

Explanation / Answer

(a)

Making use of the complement rule:

P( atleast one of the cards is a king) = 1 - P(none of the cards is a king)

P(none of the cards is a king) = (48C7)/(52C7) = 0.55

So,

P( atleast one of the cards is a king) = 1 - 0.55 = 0.45

(b)

The error made by the student is that he does not consider the possibility that there can be more than one king as well. He has calculated the probability of drawing exactly one king, while the question asks to calculate atleast one king.

Hope this helps !

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