Festivals occur once every 20 days and there is independently a 10% chance of a sale on any given day. You also know that half of previous sales have

Question

You are an art salesperson whose job is to sell paintings. You sell paintings at art festivals, as well as out of your studio on non festival days

Festivals occur once every 20 days and there is independently a 10% chance of a sale on any given day. You also know that half of previous sales have occurred during festivals.

If there is a festival what is the probability  of a sale?

This question is kinda confusing me. I did this but 0.9 doesn't seem to be right intuitively.

Would I have to look at it from the view of 20 days? or of 365 days? I am not sure about this

Answer 1

Your $0.9$ is wrong, because of your $\frac{20}{360}=\frac{1}{18}$ at the start. This would be inconsistent with "Festivals occur once every $20$ days".

You are working with

• The probability of a sale on a particular day is $P(B)=\frac1{10}$.
• The probability of a day being a festival day is $P(A)=\frac1{20}$.
• The probability of the day of a sale being a festival day is $P(A \mid B)=\frac12$.

You want to find $$P(B \mid A) = \dfrac{P(A, B)}{P(A)}=\dfrac{P(A \mid B)\, P(B)}{P(A)}=\dfrac{\frac12 \frac1{10}}{\frac1{20}}=1.$$

This makes sense: you make a sale each festival day and on average every $\frac1{19}$ of non-festival days, so on average making $2$ sales every $20$ days, i.e. on $\frac1{10}$ of days.