ProbabilityHardBloom L3

Question

From an urn containing 5 white and 5 black balls. 5 balls are transferred at random into an empty second urn from which one ball is drawn and it is found to be white. Then the probability that all balls transferred from the first urn are white is

Options

A.$\frac{1}{122}$
B.$\frac{1}{125}$
C.$\frac{1}{126}$
D.$\frac{1}{131}$

Solution

Sol. [Key Concept: Urn 5W+5B, draw 5: hypergeometric probability.]

(C)

$$\begin{matrix} P & \ = \frac{\frac{\ ^{5}C_{5}}{\ ^{10}C_{5}} \times \frac{5}{5}}{\frac{\ ^{5}C_{5}}{\ ^{10}C_{5}} \times \frac{5}{5} + \frac{\ ^{5}C_{4}\ ^{5}C_{1}}{\ ^{10}C_{5}}\frac{4}{5} + \frac{\ ^{5}C_{3}\ ^{5}C_{2}}{\ ^{10}C_{5}}\frac{3}{5} + \frac{\ ^{5}C_{2}\ ^{5}C_{3}}{\ ^{10}C_{5}}\frac{2}{5} + \frac{\ ^{5}C_{1}\ ^{5}C_{4}}{\ ^{10}C_{5}} \times \frac{1}{5}} \\ & \ = \frac{5}{5 + 100 + 300 + 200 + 25} \\ & \ = \frac{1}{126} \end{matrix}$$

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