A deck has 52 cards, including 4 aces. Two cards are drawn randomly, one by one,
ID: 3202753 • Letter: A
Question
A deck has 52 cards, including 4 aces. Two cards are drawn randomly, one by one, without replacement. Define the following events:
A: The first card is an ace.
B: Other card is an ace.
a) Find P(A), P(B), P(AB) and P(AB) (There is supposed to be a line over the last B in P(AB)).
b) Are A and B independent? Explain your answer.
c) The following game is played. You pay 1 penny (stake), and you draw two cards as described above. If both cards are aces, you win £ 200 in addition to the stake. Otherwise you win nothing and it costs you money (1 penny). Define X: net profit. Find the probability distribution of X, E (X) and Var (X).
Explanation / Answer
a) here as 4 aces are there in 52 card set ,
hence probabilty of drawing one ace out of 52 cards =P(A)=4/52=1/13
for second card to be an ace
probability =P(B)=P(first card an ace and second card an ace +first card not an ace and second card an ace)
=(4/52 *3/51+48/52*4/51) =17/221=1/13
P(AnB) =P(first card an ace and second card an ace) =(4/52 *3/51) =1/221
and P(AnB)C =1-P(AnB) =1-1/221 =220/221
b) as P(AnB) is not equal to P(A)*P(B) hence they are not independent.
c) here below is given the distribution
mean E(X) =89.59
and Var(X) =1809955.66-89.592 =1342.359
x P(x) xP(X) X^2P(X) 20000 1/221 90.50 1809954.75 -1 200/221 -0.90 0.90 total 89.59 1809955.66