Number of fours in tosses of fair die

As an illustration, let's say we throw a die, and we are interested to find the probability of obtaining a four when the die is thrown \(3\) times.

Let \(X\) be the number of fours obtained when a die is thrown \(3\) times.

In this experiment:

1. The experiment consists of \(3\) repeated trials.
2. The repeated trials are independent.
3. Each trial has two possible outcomes: either a 'success' (a four) or a 'failure' (non-four).
4. The probability of success is the same in each trial (probability of getting a four is always \(\frac{1}{6}\)).

Hence, \(X \sim B(3, \frac{1}{6})\).

\[P(\text{getting one four}) = P(X=1) = {}^{3}C_1 \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^2 = \mathrm{binompdf}\left(3, \frac{1}{6}, 1\right) \approx0.347.\]