Sure, let's work through this problem step-by-step to find the probability of getting exactly 2 heads in 3 coin flips.
### Step-by-Step Solution
1. Define the Event:
- We are interested in the event of getting exactly 2 heads in 3 coin flips.
2. Number of Trials:
- The number of trials, or coin flips, is 3.
3. Sample Space:
- Each coin flip has 2 possible outcomes: heads (H) or tails (T).
- Therefore, the total number of possible outcomes when flipping a coin 3 times is [tex]\(2^3 = 8\)[/tex].
4. Identifying Successful Outcomes:
- We need to determine the number of ways to get exactly 2 heads in 3 flips.
- This can be thought of as a combination problem where we choose 2 flips out of 3 to be heads.
- The number of combinations of 3 flips taken 2 at a time is given by the binomial coefficient [tex]\( \binom{3}{2} \)[/tex]:
[tex]\[
\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3
\][/tex]
So, there are 3 successful outcomes that result in exactly 2 heads. These outcomes are:
- HHT
- HTH
- THH
5. Calculate the Probability:
- The probability of an event is given by the ratio of the number of successful outcomes to the total number of possible outcomes.
[tex]\[
P(\text{2 heads}) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} = \frac{3}{8}
\][/tex]
6. Conclusion:
- Therefore, the probability of getting exactly 2 heads in 3 coin flips is [tex]\( \frac{3}{8} \)[/tex], which is equal to 0.375.
### Final Answer:
- The total number of possible outcomes (sample space) is 8.
- The number of successful outcomes (where we get exactly 2 heads) is 3.
- The probability [tex]\( P(\text{2 heads}) \)[/tex] is 0.375.
So, we have:
[tex]\[
\text{Total possible outcomes} = 8
\][/tex]
[tex]\[
\text{Number of successful outcomes} = 3
\][/tex]
[tex]\[
\text{Probability of getting exactly 2 heads} = 0.375
\][/tex]
This completes our step-by-step solution to the problem!
