Question:
Two cards are drawn simultaneously(or successively without replacement) from a pack of 52 cards.Find the mean and variance of the number of ashes.
Answer:
Total number of cards n = 52
Total number of ace cards = 4
Probability of getting two ace card = 4C2 / 52C2
= {(4 * 3)/(2 * 1)}/{(52 * 51)/(2 * 1)}
= (4 * 3)/(52 * 51)
= 3/(13 * 51)
= 1/(13 * 17)
= 1/221
=> p = 1/221
Now, probability of not getting two ace card q = 1 - p = 1 - 1/221 = 220/221
Now, mean = n * p = 52 * 1/221 = 52/221
Varience = n * p * q = 52 * (1/221) * (220/221) = (52 * 220)/(221 * 221) = 11440/48841
So, mean = 52/221, varience = 11440/48841
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