Question

If a fair coin is tossed 5 times, what is the probability of getting exactly 3 heads?

• A. \( \frac{5}{32} \)
• B. \( \frac{10}{32} \)
• C. \( \frac{15}{32} \)
• D. \( \frac{20}{32} \)

Hint:

For fair coins ($p=q=1/2$), the formula simplifies to $\frac{\binom{n}{r}}{2^n}$. Just calculate the combination and divide by the total power of 2!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This is a binomial probability problem. A fair coin toss is a Bernoulli trial where the probability of success (heads) $p = 1/2$ and the probability of failure (tails) $q = 1/2$.

Step 2: Key Formula or Approach
The probability of $r$ successes in $n$ trials is given by: \[ P(X = r) = \binom{n}{r} p^r q^{n-r} \] Here, $n = 5$ and $r = 3$.

Step 3: Detailed Explanation
1. Total number of outcomes = $2^n = 2^5 = 32$. 2. Number of ways to choose 3 heads out of 5 tosses: \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \] 3. Probability: \[ P(3 \text{ heads}) = \frac{10}{32} \]

Step 4: Final Answer
The probability is \( \frac{10}{32} \).