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Chapter 7: Q.7.18 (page 353)

Cards from an ordinary deck of 52playing cards are turned face upon at a time. If the 1st card is an ace, or the 2nd a deuce, or the 3rd a three, or ...,or the 13th a king,or the 14an ace, and so on, we say that a match occurs. Note that we do not require that the (13n + 1) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

Short Answer

Expert verified

The expected number of matches that occur is 4

E(N)=4

Step by step solution

01

Given Information

Let Xi be the event that when we turn over card i if matches the required cards face.

For example, X1is the event that turning over card one reveals an ace.

X2is the event that turning over the second card reveals a deuce etc.

02

Explanation 

The number of matched cards N is given by the sum of these indicator random variables as,

N=∑i=152Xi

P(Xi)=452, probability that a certain selection is a match is,

P(Xi)=452=113

E(N)=∑i=152E(Xi)

=∑i=152P(Xi)

=52.113

=4

03

Final Answer

The expected number of matches that occur is 4.

E(N)=4

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Let X1,...be independent with common mean μand common variance σ2, and set Yn=Xn+Xn+1+Xn+2. For j≥0, find CovYn,Yn+j.

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