Question

of a group of 500 people, 60 are fit enough to climb mount shasta in california. the group has 300 women and 200 men. 12% of the men are fit to climb the mountain. what is the probability that a person picked at random from the group is a male or is fit to climb mount shasta?

a. (\frac{2}{5})

b. (\frac{3}{50})

c. (\frac{13}{25})

d. (\frac{59}{125})

Explanation:

Step1: Find number of fit men

12% of 200 men are fit. So, \( 0.12 \times 200 = 24 \) fit men.

Step2: Apply principle of inclusion - exclusion

Let \( M \) be the event of being male, \( F \) be the event of being fit.
We know \( n(M) = 200 \), \( n(F) = 60 \), \( n(M \cap F)=24 \), total \( N = 500 \).
The formula for \( P(M \cup F) \) is \( \frac{n(M) + n(F)-n(M \cap F)}{N} \)
Substitute values: \( \frac{200 + 60 - 24}{500}=\frac{236}{500}=\frac{59}{125} \)? Wait, no, wait. Wait, 200 + 60 -24 = 236? Wait, no, 200+60=260, 260 -24=236? Wait, but let's recalculate. Wait, 200 (men) + 60 (fit) -24 (fit men, since they are counted twice). So 200 +60 -24 = 236. Then 236/500. Simplify: divide numerator and denominator by 4: 59/125? Wait, but let's check again. Wait, maybe I made a mistake. Wait, total people: 500. Men:200, fit:60, fit men:24. So number of people who are male or fit: men (200) + fit women (60 -24=36). So 200 +36=236. 236/500 = 59/125. Wait, but let's check the options. Option D is 59/125. Wait, but wait, maybe I messed up. Wait, let's re-express the formula. The probability of \( M \cup F \) is \( P(M) + P(F) - P(M \cap F) \). \( P(M) = 200/500 = 2/5 \), \( P(F)=60/500 = 3/25 \), \( P(M \cap F)=24/500 = 6/125 \). So \( 2/5 + 3/25 - 6/125 \). Convert to 125 denominator: \( 50/125 + 15/125 -6/125 = (50 +15 -6)/125 = 59/125 \). Yes, that's correct. So the probability is 59/125.

Answer:

D. \(\frac{59}{125}\)