Step 1: Understanding the Concept:
When a coin is tossed multiple times, the outcomes follow a binomial distribution because there are exactly two possible outcomes (success/head or failure/tail) for each independent toss.
Step 2: Key Formula or Approach:
The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
where:
$n$ = total number of trials
$k$ = number of successful trials
$p$ = probability of success on a single trial
$q$ = probability of failure on a single trial ($1 - p$)
Step 3: Detailed Explanation:
Given a fair coin tossed 5 times:
Number of trials, $n = 5$.
Desired number of heads, $k = 3$.
Probability of getting a head (success), $p = \frac{1}{2}$.
Probability of getting a tail (failure), $q = 1 - \frac{1}{2} = \frac{1}{2}$.
Plugging these values into the formula:
\[ P(X = 3) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{5-3} \]
Calculate combinations $\binom{5}{3}$:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
Calculate the powers:
\[ P(X = 3) = 10 \times \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^2 = 10 \times \left(\frac{1}{2}\right)^5 = 10 \times \frac{1}{32} = \frac{10}{32} \]
Step 4: Final Answer:
The probability of getting exactly 3 heads is $\frac{10}{32}$.