Practice Exam Fall 2019

#stat#stat416#review

Box contains 3 white and 5 black. Draw 1 ball at random from the box. Then return it + another 5 of the same color. Then draw another ball at random a. What is probability 2nd ball is white? $P(W_1{W}_2)+P(B_1{W}_2)=\frac{3\ast 8}{8\ast 13}+\frac{5\ast 3}{8\ast 13}=\frac{39}{104}$

b. If 2nd ball drawn is white, probability first ball is also white? $P(W_1|W_2)=\frac{P(W_1\cap W_2)}{P(W_2)}=\frac{\frac{3}{13}}{\frac{3}{8}}=\frac{8}{13}$

A and B roll die 10 times. Given total 3 aces between them, conditional prob that A got all aces? $P(A=3,B=0|T=3)$ or $P(A=3|A+B=3)=\frac{P(A=3,B=0)}{P(A+B=3)}=\frac{P(A=3)P(B=0)}{P(A+B=3)}$ = $\frac{\left(\genfrac{}{}{0ex}{}{10}{3}\right)(\frac{1}{6}{)}^3(\frac{5}{6}{)}^7\times \left(\genfrac{}{}{0ex}{}{10}{0}\right)(\frac{5}{6}{)}^{10}}{\left(\genfrac{}{}{0ex}{}{20}{3}\right)(\frac{1}{6}{)}^3(\frac{5}{6}{)}^{17}}=2/19$