Solution: 2018 Winter Final - 11

Author: Michiel Smid

Question

You roll a fair die 18 times; the rolls are independent of each other. What is the probability that you roll a 5 exactly three times?

(a)

None of the above.

(b)

${18 \choose 3} \cdot \left(5 \middle/ 6 \right)^{18}$

(c)

$18 \cdot \left. 5^{15} \middle/ 6^{18} \right.$

(d)

${18 \choose 3} \cdot \left. 5^{15} \middle/ 6^{18} \right.$

First, we choose which 3 of the 18 rolls are 5: $ \binom{18}{3} $

We also need to calculate the prbability of rolling a 5 exactly 3 times: ${( \frac{1}{6})}^3 $

We also need to calculate the probability of not rolling a 5 exactly 15 times: ${( \frac{5}{6})}^{15} $

$ \binom{18}{3} \cdot {( \frac{1}{6})}^3 \cdot {( \frac{5}{6})}^{15} $

$ = \binom{18}{3} \cdot ( \frac{1^3}{6^3}) \cdot ( \frac{5^{15}}{6^{15}}) $

$ = \binom{18}{3} \cdot \frac{5^{15}}{6^{18}} $