Statistics And Probability Question 82
Question: An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is
Options:
A) $ \frac{1}{10} $
B) $ \frac{3}{10} $
C) $ \frac{3}{5} $
D) $ \frac{1}{2} $
Show Answer
Answer:
Correct Answer: D
Solution:
- [d] Let $ A_{i}(i=2,3,4,5) $ be the event that urn contains $ 2,3,4,5 $ white balls and let B be the event that two white balls have been drawn than we have to find $ P(A_5/B). $ Since the four events $ A_2,A_3,A_4 $ and $ A_5 $ are equally likely we have $ P(A_2)=P(A_3)=P(A_4) $ $ =P(A_5)=\frac{1}{4}. $ $ P(B/A_2) $ is probability of event that the urn contains 2 white balls and both have been drawn.
$ \therefore P(B/A_2)=\frac{^{2}C_2}{^{5}C_2}=\frac{1}{10} $ Similarly $ P(B/A_3)=\frac{^{3}C_2}{^{5}C_2}=\frac{3}{10}, $ $ P(B/A_4)=\frac{^{4}C_2}{^{5}C_2}=\frac{3}{5},P(B/A_5)=\frac{^{5}C_2}{^{5}C_2}=1. $ By Baye?s theorem, $ P(A_5/B)=\frac{P(A_5)P(B/A_5)}{(P(A_2)P(B/A_2)+P(A_3)P(B/A_3)} $ $ +P(A_4)(B/A_4)+P(A_5)P(B/A_5)) $ $ =\frac{\frac{1}{4}.1}{\frac{1}{4}[ \frac{1}{10}+\frac{3}{10}+\frac{3}{5}+1 ]}=\frac{10}{20}=\frac{1}{2}. $
BETA
