A die is rolled five times. What is the probability that exactly two of
the results are equal to three?
We are taught to draw the trees and calculate it this way, however for a fair die being rolled five times, the diagram would be very messy. Is there another way to solve this?
Best Answer
So you're looking for 2 successes (rolling a 3) from 5 trials with a probability of success of 1/6.
\begin{equation} P(success) = {5 \choose 2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^{5-2} \end{equation}
\begin{equation} P(success) = 10\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^3 \end{equation}
P(2 threes from 5 rolls) = 0.1608 to 4 decimal places.