Question

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

Hint:

For binomial probability: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}. \]

Answer 1

Step 1: Define the Probability of Success 
A die has numbers {1,2,3,4,5,6}. The odd numbers are {1,3,5}, so the probability of success (getting an odd number) is: \[ p = \frac{3}{6} = \frac{1}{2}. \] 
Step 2: Use the Binomial Probability Formula 
The probability of exactly \( k \) successes in \( n \) trials is: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}. \] Substituting \( n = 6 \), \( k = 5 \), and \( p = \frac{1}{2} \): \[ P(X = 5) = \binom{6}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{1}. \] 
Step 3: Compute the Probability 
\[ \binom{6}{5} = \frac{6!}{5!(6-5)!} = 6. \] \[ P(X = 5) = 6 \times \left(\frac{1}{2}\right)^6. \] \[ = 6 \times \frac{1}{64}. \] \[ = \frac{6}{64} = \frac{3}{32}. \]