# Probability of three workers being in city A at the same time

### Problem

A company has 3 employees. Each of them has an independent project and must work in city A for 7 days continuously every month. For the rest of month they work in City B. What is the probability that the three of them will be in city A at same time?

They start at random dates. Also, you can assume that there are always 30 days in a month.

### Progress

I first treat the three as independent events. The probability of being in city A for each employee is $7/30$. Then I calculate at a day the probability of three people be in City A, which is $(7/30)^3$. But something is wrong, I think the probability should be greater than that.

• What are your assumptions? Are the start dates random? Do you consider varying lengths of month? What have you tried? – Ross Millikan Sep 21 '14 at 2:37
• Yes, they start dates random. Also, u can assume that there is always 30 days in a month. – junesummer0707 Sep 21 '14 at 2:40
• You have $24$ potential start days. Can you compute the chance that two are in the city together? – Ross Millikan Sep 21 '14 at 2:51
• Why do we have 24 potential start days? I first treat the three as independent event. The probability of being in city A for each employee is 7/30. Then I calculate at a day the probability of three people be in City A, which is (7/30)^3. But something is wrong, I think the probability should be greater than that. – junesummer0707 Sep 21 '14 at 2:56

Several things are wrong with your calculation. You can only start a stay on a day from the first to the 24th if the job has to get done this month. If the start days are randomly distributed (an assumption), the chance that a given employee is in the city is $\frac 1{24}$ for the first and the 30th, $frac 2{24}$ for the second and the 29th, etc. The chance that two employees are both there cannot just use these numbers because of the correlation-if they are both there on a given day the chance is high they will both be there the next day.
What you need to do is calculate the chance that when you draw three numbers (the starting dates) out of the range $[1,24]$ with replacement, the difference between the highest and lowest is $6$ or less, in which case the three will all be in the city together. It is not an easy problem.