Three cards are drawn at random (without replacement) from a well shuf

Three cards are drawn at random (without replacement) from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution.

Answer»

Let the number of red card in a sample of 3 cards drawn be random variable X. Obviously X may have values 0,1,2,3.

Now P(X = 0)= Probability of getting no red card = ²⁶C₃/⁵²C₃ = 2600/22100 = 2/17

P(X = 1)= Probability of getting one red card and two non-red cards = (²⁶C₂ x ²⁶C₂)/⁵²C₃ = 8450/22100 = 13/34

P(X = 2)= Probability of getting two red card and one non-red card = (²⁶C₂ x ²⁶C₁)⁵²C₃ = 8450/22100 = 13/34

P(X = 3)= Probability of getting 3 red cards = ²⁶C₃/⁵²C₃= 2600/22100 = 2/17

Hence, the required probability distribution in table as

X	0	1	2	3
P(X)	2/17	13/34	13/34	2/17

∴ Required mean = E(X) = ∑p_ix_i = 0 x 2/17 + 1 x 13/34 + 2 x 13/34 + 3 x 2/17

= 13/34 + 26/34 + 6/17 = (13 + 26 + 12)/34 = 51/34 = 3/2