Question

Suppose that a box contains five red balls and ten blue balls. If seven balls are selected randomly without replacement, what is the probability that at least three red balls will be obtained?

Short answer

The probability that at least three red balls will be obtained is 0.4266.

Step-by-step solution

Given information

Here X is the random variable that stands for the number of the red balls. X is following hypergeometric distribution, is \(X \sim Hyp\left( {N = 15,r = 5,n = 7} \right)\)

Therefore, the pmf of the following distribution is:

\(p\left( x \right) = \frac{{{}^5{C_x}{}^{10}{C_{7 - x}}}}{{{}^{15}{C_7}}},x = 0,1,2, \ldots \)

Calculating the probability

\(\begin{array}{l}P\left( {at\;least\;3\;balls} \right)\\ = P\left( {X \ge 3} \right)\\ = 1 - P\left( {X < 3} \right)\\ = 1 - \left[ {P\left( {X = 0} \right) + P\left( {X = 1} \right) + P\left( {X = 2} \right)} \right]\\ = 1 - \sum\limits_{x = 0}^2 {\frac{{{}^5{C_x}{}^{10}{C_{7 - x}}}}{{{}^{15}{C_7}}}} \\ \approx 0.4266\end{array}\)

Hence, the final answer is 0.4266.