Question

In a town there are 6000 people out of which 1200 are over 50 years old and 2000 are female. It is known that 30% of females are over 50 years old. What is the probability that a randomly chosen individual from the town is either female or over 50 years of age?

Answer 1

Hint: Add the number of people over 50 years of age and number of females and subtract those people from it who are counted twice, i.e., the females who are 50 years of age to get the number of favourable outcomes. The total outcomes for the above question is 6000. For finding the number of females who are older than 50 years use the point that 30% of the females are old. Finally use the formula $\text{Probability}=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ to get the answer.

Complete step by step solution:
Before moving to the question, let us talk about probability.
Probability in simple words is the possibility of an event to occur.
Probability can be mathematically defined as $=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .
Now, let’s move to the solution to the above question.
As we are asked the probability of picking a female or a person above 50 years, we will add the number of people over 50 years of age and number of females and subtract those people from it who are counted twice, i.e., the females who are 50 years of age to get the number of favourable outcomes.
Now to find the number of females who are above 50 years of age, we will use the statement that 2000 are females and 30% of which are more than 50 years old.
Females above 50 years$=\dfrac{30}{100}\times 2000=600$
Therefore, total number of favourable case$=2000+1200-600=2600$
And for the above question total outcomes is 6000.
So, the probability that a randomly chosen individual from the town is either female or over 50 years of age is $=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}=\dfrac{2600}{6000}=\dfrac{13}{30}$
Therefore, the answer to the above question is $\dfrac{13}{30}$ .

Note: Don’t get confused and take 30% of the old people to be females instead of taking 30% of the females are old, as the statement given in the question is a bit confusing. Also, don’t forget to subtract the overlapping cases, i.e., considering the possibility that there can be females who are old at the same time.