Probability

A coin is flipped 3 times. What is the probability of getting exactly 2 heads?

Event:
- Choose...

# of trials:
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[tex]\[ P(\text{2 heads}) = \][/tex]

Experiment:
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Sample Space:
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(Note: The "Choose..." options suggest that multiple choice selections were intended to be provided here. Please ensure that the necessary options are included to complete the question.)



Answer :

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!