Question

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

Answer 1

Let the random variable X= No. of red cards
Here, X can take values 0,1,2,3.
Now, P(X=0)=P(All black cards)=

\[P(B) \times P\left( BB|B \right) \times P\left( BBB|BB \right)\]

=\[\frac{26}{52} \times \frac{25}{51} \times \frac{24}{50} = \frac{2}{17}\]

 P(X=1)= \[P(RBB) + P(BRB) + P(BBR)\]

\[= 3 \times P(R) \times P(B|R) \times P(B|RB) = 3 \times \frac{26}{52} \times \frac{26}{51} \times \frac{25}{50} = \frac{13}{34}\]

P(X=3)=P(All red cards)=

\[P(R) \times P(RR|R) \times P(RRR|R) = \frac{26}{52} \times \frac{25}{51} \times \frac{24}{50} = \frac{2}{17}\]

So, the probability distribution of X is as shown:

x	0	1	2	3
P(x)	\[\frac{2}{17}\]	\[\frac{13}{34}\]	\[\frac{13}{34}\]	\[\frac{2}{17}\]

 

\[\therefore \text { Mean }, E(X) = \sum x_i p_i = 0 \times \frac{2}{17} + 1 \times \frac{13}{34} + 2 \times \frac{13}{34} + 3 \times \frac{2}{17}\]

\[ = 1 . 5\]