# Probability of Picking Groups of Cards

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Find the probability that a hand of 13 cards dealt from a well shuffled pack of 52 contains:

a) Exactly two kings and one ace

b) Exactly one ace given that it contains exactly two kings

c) Exactly 10 spades

d) At least two diamonds given that it contains exactly 3 spades and 4 hearts

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##### Solution Summary

Combinations are used to calculate the probability of picking groups of cards.

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Solution. Define C(n,k)=n!/[k!(n-k)!]. For example, C(3,2)=3!/2!=3 and C(5,2)=5!/(3!*2!)=10.

If we choose a hand of 13 cards dealt from a well shuffled pack of 52, there are C(52,13) possible cases.

(a) Exactly two kings and one ace.

There are C(4,2)*C(4,1)=6*4=24 possible case to choose exactly two kings and a ace. Then the rest of 10 cards must be chosen from those 11*4=44 cards. So there are C(44,10) possible case. So the total possible case for "exactly two kings and one ace" is ...

###### 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!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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