(2.1) For a fair five-card game, calculate the probability of having a royal flu

Question

(2.1) For a fair five-card game, calculate the probability of having a royal flush in a deal.

(2.2) If your adversary showed a royal flush in three consecutive deals, what is the probability for that to happen under the fair game assumption?

(2.3) You want to test the following hypotheses, after your adversary showed three consecutive hands of royal flush H0: The card game is fair; H1: The card game is not fair. Calculate the p-value under this setup and tell us what is your conclusion regarding the above hypothesis test.

Explanation / Answer

2.1

Probability->Number of ways of selecting the royal flush cards=4sets*1

Number of ways=52C5

Hence prob=4/52C5 =1.539x10^-6

2.2

prob of 3 consecutive royal flush=(1.539*10^-6)^3=3.645x10^-18

This probability is quite Low almost tending to 0

For any level of significance our null hypothesis is rejected

Hence we accept the alternate hypothesis that the card game is not fair.

Hence The card gave is not fair and there is enough evidence to accept the claim

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