Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success, then the probability of at most 3 successes is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{13}{16}\)
The number of successes X follows a binomial distribution with \(n = 5\) and \(p = \frac{3}{6} = \frac{1}{2}\), so
\( \displaystyle P(X = r) = {}^{5}C_{r}\left(\frac{1}{2}\right)^5 = \frac{{}^{5}C_{r}}{32} \)
"At most 3" is quickest through the complement, "4 or 5 successes":
\( \displaystyle P(X \le 3) = 1 - P(X = 4) - P(X = 5) \)
\( \displaystyle = 1 - \frac{5}{32} - \frac{1}{32} = \frac{26}{32} = \frac{13}{16} \)