Question:
You have obtained some interesting information about the local lottery. There was a malfunction at the printer that accidentally marked a bunch of tickets with a red dot. This malfunction disproportionately affected winning lottery tickets. In total $40\%$ of winning tickets were marked with a red dot, while only $20\%$ of losing tickets were marked with a red dot. You have a probability of $\frac{3}{39}$ of winning the lottery.
You have found a ticket marked with a red dot. What is the probability that this is a winning ticket?
What i have done is the following:
A= probability of a winning ticket.
B= probability of having a red dot.
P(A|B)= P(A intersection B) / P(B)
P(B)= P(red dot| winning ticket)$\times$ P(winning ticket) + P(red dot| losing ticket)$\times$ P(losing ticket)
$0.4 \times \frac{3}{39} + 0.20 \times \frac{12}{13}= 0.215$
P(A intersection B)= P(A|B)$\times$P(B)= $0.4\times 0.215=0.086$
So P(A|B)= $\frac{0.086}{0.215}=0.4$
I'm not sure if I'm doing this right. Maybe someone can give feedback.
Best Answer
Out of $39$ tickets, $3$ are winning tickets and $36$ are losing tickets.
So, number of winning tickets with red dot = $3\times 0.4$
Number of losing tickets with red dot = $36\times 0.2$
You need to find probability of winning of a ticket with red dot = $\dfrac{3\times 0.4}{3\times 0.4+36\times 0.2} = \dfrac{1}{7}$