Final answer:
The overall probability that a person chosen at random from the social club does not wear glasses is calculated by finding the respective probabilities for men and women and then combining them. After performing the calculations, the probability is found to be 52/79.
Step-by-step explanation:
To find the probability that a person chosen at random from the social club does not wear glasses, we need to calculate the probability separately for men and women and then combine them using the total number of people in the club as the basis.
For men, the probability that a man does not wear glasses is 1 - 2/9 = 7/9. Since there are 54 men, the number of men who don't wear glasses is 54 * 7/9.
For women, the probability that a woman does not wear glasses is 1 - 3/5 = 2/5. There are 25 women, so the number of women who don't wear glasses is 25 * 2/5.
Adding both groups together gives us the total number of people who don't wear glasses. To find the overall probability, we divide this sum by the total number of people in the club. Therefore, the overall probability that a randomly chosen person does not wear glasses is ((54 * 7/9) + (25 * 2/5)) / (54 + 25).
Performing the arithmetic, we get:
- For men: 54 * 7/9 = 42
- For women: 25 * 2/5 = 10
Total number of people who don't wear glasses = 42 + 10 = 52
Total number of people in the club = 54 + 25 = 79
Finally, the overall probability is 52/79.