Two sixes in six rolls

Problem

A fair die is rolled $6$ times. What is the probability of getting the number $6$ exactly $2$ times?

Solution

Each roll is a Bernoulli trial with «success» = a $6$ comes up, probability $p=1/6$. Over $n=6$ rolls the number of successes follows the binomial scheme: $P(k=2) = \binom{6}{2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^4 = 15\cdot\frac{1}{36}\cdot\frac{625}{1296} \approx \boxed{0,2}$

Links

Topics: Probabilita
Concepts: Distribuzione binomiale
Methods: Binomiale
Skills: Calcolo probabilita
Exercise type: Problema probabilita