2) Suppose that a card player will be dealt a random subset of 5 cards twenty di

Question

2) Suppose that a card player will be dealt a random subset of 5 cards twenty different times, each time starting with a randomly-ordered standard deck of 52 playing cards. What is the expected number of times he'll get at least two aces among his five cards? at an ordinary six-sided fair die will be rolled until 6 spots appear on its upward face, and let e number of rolls required. Letting X be a normal random variable having mean 5 and variance 3, value of P(X > EY), rounding your answer to the nearest thousandth. :, ( 1 lhe probab:),of succeu on each trial i,

Explanation / Answer

2.

total playing cards = 52( total Aces =4)
each time can take 5 cards randomly from pack of cards
no of times = 20
probability = 5/52 = 0.096

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials   
n = is the number of independent trials   
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 20 * 0.096
= 1.92
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 20 * 0.096 * 0.904
= 1.7357
III.
standard deviation = sqrt( variance ) = sqrt(1.7357)
=1.3175

Expected no of times he will get atleast 2 ace cards among his 5 cards
P( X < 2) = P(X=1) + P(X=0)
= ( 20 1 ) * 0.096^1 * ( 1- 0.096 ) ^19 + ( 20 0 ) * 0.096^0 * ( 1- 0.096 ) ^20   
= 0.415
P( X > = 2 ) = 1 - P( X < 2) = 0.585

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