To solve this problem, we need to find the overall probability that a randomly chosen person from the running club has blue eyes. We will achieve this by considering the probabilities for both females and males having blue eyes, and then combining these probabilities appropriately.
### Step-by-step Solution:
1. Determine the total number of people in the club:
- There are [tex]\( 50 \)[/tex] females and [tex]\( 80 \)[/tex] males.
- Thus, the total number of people is:
[tex]\[
50 + 80 = 130
\][/tex]
2. Calculate the probability of selecting a female:
- There are [tex]\( 50 \)[/tex] females out of [tex]\( 130 \)[/tex] total people.
- Therefore, the probability of selecting a female ( [tex]\( P(\text{Female}) \)[/tex] ) is:
[tex]\[
P(\text{Female}) = \frac{50}{130} = \frac{5}{13}
\][/tex]
3. Calculate the probability of selecting a male:
- There are [tex]\( 80 \)[/tex] males out of [tex]\( 130 \)[/tex] total people.
- Therefore, the probability of selecting a male ( [tex]\( P(\text{Male}) \)[/tex] ) is:
[tex]\[
P(\text{Male}) = \frac{80}{130} = \frac{8}{13}
\][/tex]
4. Determine the probability of a female having blue eyes:
- The probability that a female has blue eyes is given as [tex]\( 0.38 \)[/tex].
5. Determine the probability of a male having blue eyes:
- The probability that a male has blue eyes is given as [tex]\( 0.6 \)[/tex].
6. Calculate the total probability that a person has blue eyes:
- This can be found by considering the probability of selecting a female and the probability she has blue eyes, and the probability of selecting a male and the probability he has blue eyes.
- Using the law of total probability, we get:
[tex]\[
P(\text{Blue Eyes}) = P(\text{Female}) \cdot P(\text{Blue Eyes}|\text{Female}) + P(\text{Male}) \cdot P(\text{Blue Eyes}|\text{Male})
\][/tex]
- Substituting the known values:
[tex]\[
P(\text{Blue Eyes}) = \left(\frac{5}{13}\right) \cdot 0.38 + \left(\frac{8}{13}\right) \cdot 0.6
\][/tex]
- Simplifying these:
[tex]\[
P(\text{Blue Eyes}) = \left(\frac{5}{13}\right) \cdot 0.38 + \left(\frac{8}{13}\right) \cdot 0.6 = \frac{5 \times 0.38 + 8 \times 0.6}{13}
\][/tex]
- Further simplification gives:
[tex]\[
P(\text{Blue Eyes}) = \frac{1.9 + 4.8}{13} = \frac{6.7}{13} \approx 0.5154
\][/tex]
### Conclusion:
- The probability that the person chosen at random from the running club has blue eyes is approximately [tex]\( 0.5154 \)[/tex].
- Since [tex]\( 0.5154 \)[/tex] is greater than [tex]\( 0.5 \)[/tex], we can conclude that the probability the person has blue eyes is indeed more than [tex]\( 0.5 \)[/tex].
Thus, the mathematical evidence supports that the probability of selecting a person with blue eyes from the club is more than [tex]\( 0.5 \)[/tex].
