# Interesting probability question

Your local art museum has planned next year’s $52$-week calendar by scheduling a mixture of $1$-week and 2-week shows that feature the works of $22$ painters and $20$ sculptors. There is a showing scheduled for every week of the year, and only one artist is featured at a time. There are $42$ different shows scheduled for next year.

You have randomly selected one week to attend and have been told the probability of it being a $2$-week show of sculpture is $3/13$.

a. What is the probability that the show you have selected is a painter’s showing?

b. What is the probability that the show you have selected is a sculptor’s showing?

c. What is the probability that the show you have selected is a 1-week show?

d. What is the probability that the show you have selected is a 2-week show?

This is where I got stuck. I multiplied $\frac{3}{13}$ by 52 to obtain 12, which I took to mean that 6 sculptors would do two-week shows. It didn't initially occur to me that this meant 14 sculptors had to do one-week shows, and that 26 out of the 52 weeks of the year would be devoted to sculpting. It's rather difficult to describe where I initially had difficulties, given that I'm being asked to do so after seeing two eloquent solutions to the problem, but as I said I got as far as determining that there would be six, two-week sculptors and was not sure how to proceed from there. I hope that this brief addition to the problem rectifies any nonconformity issues, as pointed out by members of this forum.

• I initially felt there was not enough info to calculate the requested probabilities. I multiplied 52 by 3/13 to obtain 12 and understood that this meant that 6 sculptors would do two-week shows, but hit a mental block at this point. I would like to sincerely thank the two individuals who provided detailed solutions to the problem. Commented Mar 19, 2014 at 16:13
• @Did, Please let me know if the previous comment will allow you to lift the hold status, or is there something else I need to do. I am relatively new to this exchange, and am not yet familiar with the requirements to which my questions must conform. Commented Mar 19, 2014 at 20:13
• to facilitate the re-opening process, you should always make additions to the question by editing it (click on the "edit" link below your question) instead of posting comments. These edits will insert your question into the review queue for community members to evaluate. (Just commenting won't do that.) Commented Mar 20, 2014 at 8:39
• Thank you Mr. Wong. I added some details to the problem, which I hope will resolve any nonconformity issues. Commented Mar 22, 2014 at 11:01

There is a $\frac{3}{13}=\frac{12}{52}$ chance of a sculptor doing a two week show, so 6 sculptors are doing 2 weeks show, meaning 14 are doing a 1 week show. Thus 14+12=26 weeks are dedicated to sculptors. 52-26=26, thus 26 weeks are dedicated to painters.

A)$\frac{26}{52}= .50$ there is a 50% chance a painter is shown.

B) a 50% chance a sculptor is showing. Now there are 22 painters for 26 weeks, meaning 4 have to do two week shows.

C) 14 sculptors and 18 painters are doing one week shows. 14+18=32. $\frac{32}{52}$ is the probability of a one week show.

D)$\frac{20}{52}$ chance of a two week show.

• Many thanks Fluke_of_Luke and Graham Kemp. Commented Mar 19, 2014 at 10:18

There must be 10 two-week shows, and 32 one-week shows. $P(W_1) = \frac{32}{52} = \frac{8}{13}, P(W_2)=\frac{20}{52}=\frac{5}{13}$

From the given information [$P(S\cap W_2)=\tfrac{3}{13}$] then:

$$P(S \mid W_2) = \frac{P(S \cap W_2)}{P(W_2)} = \frac{3}{13}\times\frac{13}{5} = \frac{3}{5}$$ $$\therefore P(P \mid W_2) = 1-P(S \mid W_2) = \frac{2}{5}$$

Thus of the ten two-week shows, six are sculpters, and four are painters. Which means that of the thirty-two one-week shows, fourteen are sculptors, and eighteen are painters.

$$P(S) = P(S \mid W_1)P(W_1)+P(S\cap W_2) = \frac{14}{32}\times \frac{8}{13}+\frac{3}{13} = \frac{13}{26} = \frac{1}{2}$$

$$P(P) = 1-P(S) = \frac{1}{2}$$

• a. What is the probability that the show you have selected is a painter’s showing? $\dfrac{1}{2}$
• b. What is the probability that the show you have selected is a sculptor’s showing? $\dfrac{1}{2}$
• c. What is the probability that the show you have selected is a 1-week show? $\dfrac{8}{13}$
• d. What is the probability that the show you have selected is a 2-week show?$\dfrac{5}{13}$