A standard deck of 52 cards is shuffled and four cards are drawn at random (with

Question

A standard deck of 52 cards is shuffled and four cards are drawn at random (without replacement). Let X be the number of aces drawn. Note that a standard deck has 4 aces. We might consider each "ace" a success and each of the four cards drawn a "trial, " but nevertheless X is not binomial. Why? Suppose a pack of Pokemon trading cards has a rare card inside with probability 0.04, independently of other packs. You plan to purchase and open packs until you get a rare card. Let Y be the number of packs of cards you purchase and open. Find a suitable distribution for Y. How many packs can you expect to buy? What is the probability you will get a rare card by the twentieth pack? Given that you have already purchased and opened at least 10 packs (and not gotten a rare card), what is the probability you will open at least 30 packs total? (You must use the conditional probability formula.)

Explanation / Answer

5) as without replacement, probabilty of drawing ace does change with each draw. Therefore probabilty of drawing an ace is not independent which violate condition of independence for binomial.

6)a) it is geometric distribution with parameter p=0.04

b) expected value =1/p=1/0.04=25

c)probability of getting a rare card by 20th pack=P(not getting card till 19th pack*getting rare card in 20th pack)

=(1-0.04)19*0.04 =0.018417

d)here probabilty P(X>30|X>10) =(1-0.04)30/(1-0.04)10 =(1-0.04)20 =0.442002

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