# A child rolls a 6-sided die 6 times. What is the probability of the child rolling exactly three sixes?

Sep 15, 2017

Pr(Three 6's) = $\frac{1}{216}$

#### Explanation:

Assuming the die is fair, on the first roll, the child has a $\frac{1}{6}$ chance of getting any number, including 6. On the next roll, the child has a $\frac{1}{36}$ chance of getting any two numbers, including two 6's. This is because the rolls are independent, so the total possible outcomes multiply each roll with each roll's probability. Since each roll's probability is $\frac{1}{6}$,
$\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$

Finally, the same thing happens again on the third roll, the probabilities multiply once more, so
$\frac{1}{36} \cdot \frac{1}{6} = \frac{1}{216}$, meaning there is a $\frac{1}{216}$ chance of getting any possible combination of three numbers from 1-6. This includes getting three 6's, therefore the probability of getting three 6's is $\frac{1}{216}$

I hope that helped!