To determine the probability that at most 4 of the selected serves are successful, we need to sum the probabilities of 0, 1, 2, 3, and 4 successful serves, as given in the table.
Given probabilities for the number of successful serves are:
- 0 successes: [tex]\(0.0002\)[/tex]
- 1 success: [tex]\(0.004\)[/tex]
- 2 successes: [tex]\(0.033\)[/tex]
- 3 successes: [tex]\(0.132\)[/tex]
- 4 successes: [tex]\(0.297\)[/tex]
The probability that at most 4 serves are successful (i.e., [tex]\(P(X \leq 4)\)[/tex]) is calculated by summing these probabilities:
[tex]\[
P(X \leq 4) = P(0) + P(1) + P(2) + P(3) + P(4)
\][/tex]
[tex]\[
P(X \leq 4) = 0.0002 + 0.004 + 0.033 + 0.132 + 0.297
\][/tex]
Adding these together:
[tex]\[
0.0002 + 0.004 + 0.033 + 0.132 + 0.297 = 0.4662
\][/tex]
Therefore, the probability that at most 4 of the serves were successful is [tex]\(0.4662\)[/tex].
Looking at the provided multiple-choice options:
- [tex]\(0.297\)[/tex]
- [tex]\(0.466\)[/tex]
- [tex]\(0.534\)[/tex]
- [tex]\(0.822\)[/tex]
The closest option to our calculated probability [tex]\(0.4662\)[/tex] is [tex]\(0.466\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{0.466}
\][/tex]
