let A denote the event that the chosen individual is female and B denote the event that the chosen individual is over 50 years old.
Given : Town consists of 6000 people, 1200 are over 50 years old, and 2000 are females
To find : Probability that a randomly chosen individual from the town is either female or over 50 years = P(A or B)
The formula used : Probability =
= \(\frac{favourable\,number\,of\,outcomes}{Total\,no.of\,outcomes}\)
P(A or B) = P(A) + P(B) - P(A and B)
For the event A ,
There are 2000 females present in a town of 6000 people
Favourable number of outcomes = 2000
Total number of outcomes = 6000
P(A) = \(\frac{2000}{6000}\) = \(\frac{1}{3}\)
For the event B,
There are 1200 are over 50 years of age in a town of 6000 people
Favourable number of outcomes = 1200
Total number of outcomes = 6000
P(A) = \(\frac{2000}{6000}\) = \(\frac{1}{5}\)
30% of the females are over 50 years
For the event A and B,
\(\frac{30}{100}\times2000 = 600\) females are over 50
Favourable number of outcomes = 600
P(A and B) = \(\frac{600}{6000}\) = \(\frac{1}{10}\)
= P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = \(\frac{1}{3}+\frac{1}{5}+\frac{1}{10}\)
P(A or B) = \(\frac{10+6-3}{30}\) = \(\frac{13}{30}\)
P(A or B) = \(\frac{13}{30}\)
The probability that a randomly chosen individual from the town is either female or over 50 years = P(A or B) = \(\frac{13}{30}\)
