Shuffle a standard deck of cards. Find the following. (a) The expected number of

Question

Shuffle a standard deck of cards. Find the following. (a) The expected number of jacks in the first five cards. (b) The expected number of jacks before the first queen. (c) The expected number of jacks before the second queen. (d) The expected number of jacks before the "Q Heart" Shuffle a standard deck of cards. Find the following. (a) The expected number of jacks in the first five cards. (b) The expected number of jacks before the first queen. (c) The expected number of jacks before the second queen. (d) The expected number of jacks before the "Q Heart"

Explanation / Answer

(a)The expected number of jacks in the first five cards can be foundas follows:

X(no. of jacks) 0 1 2 3 4 P(X) 0.658841998 0.299473636 0.039929818 0.001736079 0.000018 XP(X) 0 0.299473636 0.079859636 0.005208237 0.000074 hence expected no. of jacks = XP(X) = 0.384615

(b)The expected number of jacks before the first queen.
X(no. of jacks) 0 1 2 3 4 P(X) 0.208931265 0.081534152 0.008735802 0.000270878 0.000002 XP(X) 0 0.081534152 0.017471604 0.0008126 0.0000062
hence expected no. of jacks = XP(X) = 0.099825

(c)The expected number of jacks before the second queen.if 5 cards aredrawn, then there can be maximum of 3 jacks before the 2ndqueen.
hence,
X(no. of jacks) 0 1 2 3 P(X) 0.030575307 0.008735802 0.000609475 0.000009 XP(X) 0 0.008735802 0.001218949 0.0000277
hence expected no. of jacks = XP(X) = 0.00998


(d)The expected number of jacks before the "Q Heart"
X(no. of jacks) 0 1 2 3 4 P(X) 0.052232816 0.020383538 0.002183951 6.77194E-05 0.000000 XP(X) 0 0.020383538 0.004367901 0.0002032 0.0000015
hence expected no. of jacks = XP(X) =0.024956
X(no. of jacks) 0 1 2 3 4 P(X) 0.658841998 0.299473636 0.039929818 0.001736079 0.000018 XP(X) 0 0.299473636 0.079859636 0.005208237 0.000074

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