Question

Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least 3 successes.

Answer 1

Let X denote the occurrence of 3,4 or 5 in a single die. Then, X follows binomial distribution with n=5.
Let p=probability of getting 3,4, or 5 in a single die .
p =  \[\frac{3}{6} = \frac{1}{2}\]

\[q = 1 - \frac{1}{2} = \frac{1}{2} \]
\[P(X = r) = ^{5}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{5 - r} \]
\[P(\text{ at least 3 successes } ) = P(X \geq 3) \]
\[ = P(X = 3) + P(X = 4) + P(X = 5)\]
\[ = ^{5}{}{C}_3 \left( \frac{1}{2} \right)^3 \left( \frac{1}{2} \right)^{5 - 3} + ^{5}{}{C}_4 \left( \frac{1}{2} \right)^4 \left( \frac{1}{2} \right)^{5 - 4} +^{5}{}{C}_5 \left( \frac{1}{2} \right)^5 \left( \frac{1}{2} \right)^{5 - 5} \]
\[ = \frac{^{5}{}{C}_3 + ^{5}{}{C}_4 + ^{5}{}{C}_5}{2^5}\]
\[ = \frac{1}{2}\]