An urn contains 50 black balls and 20 blue balls.If you select 30 balls at random,what is the probability that exactly 20 of them will be black balls?
A)0.2581
B)0.3861
C)0.6667
D)0.1573
E)0.4770

Question

Quizplus · Accepted Answer

This probability can be calculated using the hypergeometric distribution formula, which is appropriate for problems of this type where you are drawing without replacement. The formula is: $$ P(X=k) = \frac{{\binom{K}{k} \cdot \binom{N-K}{n-k}}}{{\binom{N}{n}}} $$Where:- $N$ is the total number of balls (70),- $K$ is the total number of black balls (50),- $n$ is the total number of balls drawn (30),- $k$ is the number of black balls drawn (20),- $\binom{a}{b}$ is the binomial coefficient, representing the number of ways to choose $b$ items out of $a$ items without regard to order.Plugging in the numbers:$$ P(X=20) = \frac{{\binom{50}{20} \cdot \binom{20}{10}}}{{\binom{70}{30}}} $$Calculating this gives a probability of 0.1573, which corresponds to choice D.

An Urn Contains 50 Black Balls and 20 Blue Balls