Question

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.

Hint:

For binomial probabilities, use the binomial distribution formula and sum the probabilities for the required number of successes.

Answer 1

Step 1: The probability of getting an odd number on a die is \( P(\text{odd}) = \frac{3}{6} = \frac{1}{2} \). Step 2: The number of successes (odd numbers) follows a binomial distribution with \( n = 6 \) trials and \( p = \frac{1}{2} \) probability of success. We want to find \( P(\text{at least 5 successes}) \), which is: \[ P(X \geq 5) = P(X = 5) + P(X = 6) \] Step 3: Use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \] Step 4: Calculate \( P(X = 5) \) and \( P(X = 6) \): \[ P(X = 5) = \binom{6}{5} \left( \frac{1}{2} \right)^5 \left( \frac{1}{2} \right)^1 = 6 \times \frac{1}{32} = \frac{6}{32} = \frac{3}{16} \] \[ P(X = 6) = \binom{6}{6} \left( \frac{1}{2} \right)^6 = 1 \times \frac{1}{64} = \frac{1}{64} \] Step 5: Add the probabilities: \[ P(X \geq 5) = \frac{3}{16} + \frac{1}{64} = \frac{12}{64} + \frac{1}{64} = \frac{13}{64} \] Thus, the probability of at least 5 successes is: \[ \boxed{\frac{13}{64}} \]