A die is thrown 7 times. What is the chance that an odd number turns up (i) exactly 4 times (ii) at least 4 times

Explanation

To solve the problem, we will use the binomial probability formula. The probability of getting an odd number when a die is thrown is \( p = \frac{3}{6} = \frac{1}{2} \) and the probability of not getting an odd number is \( q = 1 - p = \frac{1}{2} \).

### Step-by-Step Solution

**Step 1: Identify the parameters**
- The die is thrown \( n = 7 \) times.
- The probability of getting an odd number (success) is \( p = \frac{1}{2} \).
- The probability of not getting an odd number (failure) is \( q = \frac{1}{2} \).
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