To find the probability of getting exactly 4 bull's-eyes when an archer shoots 10 arrows, where the probability of hitting a bull's-eye (success) is \( p = 0.53 \), we can use the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]
where:
- \( n \) is the total number of trials (arrows shot),
- \( k \) is the number of successes (bull's-eyes),
- \( p \) is the probability of success on each trial,
- \( \binom{n}{k} \) is the binomial coefficient.
In this case,
- \( n = 10 \)
- \( k = 4 \)
- \( p = 0.53 \)
- \( 1 - p = 0.47 \)
First, calculate the binomial coefficient \( \binom{10}{4} \):
\[ \binom{10}{4} = \frac{10!}{4!(10 - 4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
Next, calculate the probability:
\[ P(X = 4) = \binom{10}{4} (0.53)^4 (0.47)^{10 - 4} \]
Calculating \( (0.53)^4 \):
\[ (0.53)^4 \approx 0.0788 \]
Calculating \( (0.47)^6 \):
\[ (0.47)^6 \approx 0.0462 \]
Now put it all together:
\[ P(X = 4) = 210 \times 0.0788 \times 0.0462 \approx 210 \times 0.00364176 \approx 0.765 \]
Hence,
\[ P(X = 4) \approx 0.179 \]
Finally, the probability that the archer hits exactly 4 bull's-eyes when shooting 10 arrows is approximately \( 0.179 \).
So, the answer is:
0.179
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