First, let's extract the relevant information from the given probability distribution table. We know the following probabilities:
- The probability that the gymnast scores a 5 is [tex]\( P(X = 5) = 0.02 \)[/tex]
- The probability that the gymnast scores a 5.5 is [tex]\( P(X = 5.5) = 0.02 \)[/tex]
- The probability that the gymnast scores a 6 is [tex]\( P(X = 6) = 0.06 \)[/tex]
- The probability that the gymnast scores a 6.5 is [tex]\( P(X = 6.5) = 0.1 \)[/tex]
- The probability that the gymnast scores a 7 is [tex]\( P(X = 7) = 0.16 \)[/tex]
- The probability that the gymnast scores a 7.5 is [tex]\( P(X = 7.5) = 0.14 \)[/tex]
- The probability that the gymnast scores an 8 is [tex]\( P(X = 8) = 0.18 \)[/tex]
- The probability that the gymnast scores an 8.5 is [tex]\( P(X = 8.5) = 0.18 \)[/tex]
- The probability that the gymnast scores a 9 is [tex]\( P(X = 9) = 0 \)[/tex]
- The probability that the gymnast scores a 9.5 is [tex]\( P(X = 9.5) = 0.02 \)[/tex]
The objective is to find the probability that the gymnast scores either a 9 or a 9.5. To do this, we can simply add together the individual probabilities for these two specific scores:
[tex]\[
P(X = 9 \text{ or } X = 9.5) = P(X = 9) + P(X = 9.5)
\][/tex]
From the information provided:
[tex]\[
P(X = 9) = 0
\][/tex]
[tex]\[
P(X = 9.5) = 0.02
\][/tex]
Adding these probabilities together, we get:
[tex]\[
P(X = 9 \text{ or } X = 9.5) = 0 + 0.02 = 0.02
\][/tex]
Thus, the probability that the gymnast scores a 9 or a 9.5 is:
[tex]\[
\boxed{0.02}
\][/tex]
