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Probability Using Cards

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Question: Suppose we turn over cards simultaneously from two well shuffled decks of ordinary playing cards. We say we obtain an exact match on a particular turn if the same card appears from each deck: for example, the queen of spades against the queen of spades. Let pm equal the probability of at least one exact match.

Show that:
pm= 1-(1/2!)+ (1/3!) - (1/4!) +... - (1/52!).
Hint: Let Ci denote the event of an exact match on the ith turn. Then pm= P(C1 U C2 U...U C52). Now use the general inclusion-exclusion formula. Note that: P(Ci)= 1/52 and hence p1= 52(1/52) = 1. Also P(Ci intersect Cj)= 50!/52! and, hence, p2= (52 choose 2)/ (52*51).

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This solution is comprised of a very detailed, step by step response which works through this statistical problem. In order to view the solution, a Word document needs to be opened.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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