Problem Statement
A die is thrown $6$ times. If 'getting an odd number' is a success, what is the probability of:
(i) At least $5$ successes?
(ii) At most $5$ successes?
Verified Solution & Marking Scheme
Model as Binomial Experiment
Number of trials $n = 6$.
Probability of success (odd number: 1, 3, 5) $p = \frac{3}{6} = \frac{1}{2}$.
Probability of failure $q = 1 - p = \frac{1}{2}$.
$P(X = r) = \binom{6}{r} \left(\frac{1}{2}\right)^6 = \frac{\binom{6}{r}}{64}$
Part (i): At Least 5 Successes
$P(X \ge 5) = P(X = 5) + P(X = 6) = \frac{\binom{6}{5} + \binom{6}{6}}{64} = \frac{6 + 1}{64} = \frac{7}{64}$
Part (ii): At Most 5 Successes
$P(X \le 5) = 1 - P(X = 6) = 1 - \frac{1}{64} = \frac{63}{64}$