To determine the probability of getting exactly 3 heads when a coin is flipped 8 times, we can follow a clear set of steps.
### Step 1: Calculate the Total Sample Space ([tex]\( n(S) \)[/tex])
When a coin is flipped, there are 2 possible outcomes: heads (H) or tails (T). The total number of outcomes when flipping a coin 8 times can be found by raising the number of outcomes for one flip (which is 2) to the power of the number of flips (which is 8). This is calculated as:
[tex]\[
n(S) = 2^8 = 256
\][/tex]
### Step 2: Calculate the Number of Desired Outcomes ([tex]\( n(A) \)[/tex])
Next, we need to determine the number of ways to get exactly 3 heads out of 8 flips. This can be found using combinations, as the order in which the heads and tails appear does not matter. The combination formula is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
where [tex]\( n \)[/tex] is the total number of flips, and [tex]\( k \)[/tex] is the number of heads. Plugging in our values, we get:
[tex]\[
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}
\][/tex]
Calculating this, we'll find:
[tex]\[
\binom{8}{3} = 56
\][/tex]
So, the number of desired outcomes ([tex]\( n(A) \)[/tex]) is 56.
### Step 3: Calculate the Probability ([tex]\( P(A) \)[/tex])
Finally, we use the formula [tex]\( P(A) = \frac{n(A)}{n(S)} \)[/tex] to find the probability of getting exactly 3 heads in 8 flips. Substituting the values we found:
[tex]\[
P(A) = \frac{n(A)}{n(S)} = \frac{56}{256}
\][/tex]
To find the probability, we can divide and then round to the nearest hundredth:
[tex]\[
P(A) = 0.21875 \approx 0.22
\][/tex]
### Summary
After following these steps, the results are:
- Total sample space ([tex]\( n(S) \)[/tex]): 256
- Number of desired outcomes ([tex]\( n(A) \)[/tex]): 56
- Probability ([tex]\( P(A) \)[/tex]): 0.22 (rounded to the nearest hundredth).
So, the probability of getting exactly 3 heads when flipping a coin 8 times is 0.22.
