(2) Suppose you keep drawing a card (without looking) from a well-shuffled deck

Question

(2) Suppose you keep drawing a card (without looking) from a well-shuffled deck of 52 cards, each time returning the card to the deck before drawing again. You stop as soon as you get your first king. What are the chances that you have to (a) draw 20 times to get the first king? (b) draw more than 35 times to get the first king? (c) draw at most 10 times to get the first king? (3) In the problem above, if X - the number of failed draws BEFORE you get the first re' means not getting a king), what are the mean and the variance of x? king (failure' means not gett

Explanation / Answer

2)probability to draw a king =4/52 =1/13 (as there are 4 king in a 52 deck card)

a) probability to get first king on 20th draw=P(till first 19 draw no king and on 20th draw gets the king)

=(1-1/13)19*(1/13) =0.0168

b) Probability=P(till 35 draw we get no king) =(1-1/13)35 =0.0607

c)

probability =1-P(till 10 draws no king) =1-(1-1/13)10 =1-0.4491 =0.5509

3)

here radom variable Y, number of draws to get first king follows geometeric distribution with

mean E(Y)=1/p=1/(1/13)=13

and variance =(1-p)/p2 =156

here as X=Y-1

therefore mean of X =E(X)=E(Y)-1 =13-1=12

and Variance of X =Var(X)=Var(Y)=156

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