Answer to Question #209856 in Statistics and Probability for niks

Question

Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. Find the probability distribution of the number of black balls. 

Answer 1

Number of Red balls $=4$  

Number of Black balls $=3$

Total number of balls $=4+3=7$

Given: two balls are drawn in succession without replacement. 

Let $X=$ the number of black balls: $X=0,1,2.$

$$P(X=0)=\dfrac{\dbinom{4}{2}\dbinom{3}{0}}{\dbinom{7}{2}}=\dfrac{6}{21}=\dfrac{2}{7}$$

$$P(X=1)=\dfrac{\dbinom{4}{1}\dbinom{3}{1}}{\dbinom{7}{2}}=\dfrac{4(3)}{21}=\dfrac{4}{7}$$

$$P(X=2)=\dfrac{\dbinom{4}{0}\dbinom{3}{2}}{\dbinom{7}{2}}=\dfrac{3}{21}=\dfrac{1}{7}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 \\ \hline p(x) & 2/7 & 4/7 & 1/7 \\ \end{array}$$

$$E[X]=0(\dfrac{2}{7})+1(\dfrac{4}{7})+2(\dfrac{1}{7})=\dfrac{6}{7}$$