If you flip a coin 3 times, and find the probability of getting exactly two heads, this is compound probability using both "and" and "or" since it is

Understanding the Problem

To find the probability of getting exactly two heads when flipping a coin three times, we can use binomial probability. Each flip of the coin is an independent event with two possible outcomes: heads (H) or tails (T).

Relevant Concepts

1. Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where:
   • $n$ = total number of trials (in this case, 3 flips)
   • $k$ = number of successful outcomes (for this problem, 2 heads)
   • $p$ = probability of success on an individual trial (for a fair coin, $p = 0.5$)
   • $\binom{n}{k}$ = combination of $n$ taken $k$ at a time

Calculation Steps

1. Identify Variables:

   • $n = 3$
   • $k = 2$
   • $p = 0.5$

2. Calculate the Combination: $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3$

3. Compute the Probability: $P(X = 2) = \binom{3}{2} (0.5)^2 (0.5)^{3-2} = 3 \times (0.5)^2 \times (0.5)^1$ $= 3 \times 0.25 \times 0.5 = 3 \times 0.125 = 0.375$

Verify and Summarize

The calculations lead us to the conclusion that the probability of getting exactly two heads when flipping a coin three times is $0.375$ or $37.5\%$. This encompasses the independent events of obtaining heads and tails in the flips, adhering to the principles of compound probability.

Final Answer

The probability of getting exactly two heads in three coin flips is $\frac{3}{8}$ or $0.375$.