Solutions to Probability Puzzle: Find 3 Emergency Trained Staff

[statsluke]

Any possible solutions to the following probability puzzler? Cheers

A site with a rotating workforce provides emergency response training for some regular members of staff. In an emergency they are called into the emergency team. On the site 20% have emergency training. I want to find 3 emergency trained staff, and the contact list is out of date. What is the probability of getting 3 emergency trained staff after 3 phone calls. What is the probability of getting 2 after 3 calls?

 

[honestrosewater]

You don't even know it there are 3 emergency-trained staff. What if there's only 5 staff total? How many people are there at one time? How many of them are emergency-trained?
You said only the emergency-trained go on a call, so the question doesn't really make sense. Or do you just mean you're calling the place and want to know if the person who answers the phone is emergency-trained?

 

[tacman]

There are a few possible solutions to this probability puzzle. One solution is to simply make three phone calls to any randomly selected members of the staff and hope that at least three of them are emergency trained. However, this solution does not take into account the fact that the contact list is out of date and may not accurately reflect the current emergency training status of the staff members.

Another solution could be to first update the contact list by contacting all 20% of the staff members who are known to have emergency training. This would ensure that the list is accurate before making any phone calls. Then, the probability of getting 3 emergency trained staff after 3 phone calls would be calculated as follows:

(0.2)(0.2)(0.2) = 0.008 or 0.8%

This means that there is a 0.8% chance of getting 3 emergency trained staff after 3 phone calls. Similarly, the probability of getting 2 emergency trained staff after 3 phone calls would be:

(0.2)(0.2)(0.8) + (0.2)(0.8)(0.2) + (0.8)(0.2)(0.2) = 0.048 or 4.8%

This means that there is a 4.8% chance of getting 2 emergency trained staff after 3 phone calls.

Another solution could be to prioritize the phone calls by first contacting the 20% of staff members who are known to have emergency training, followed by randomly selecting from the remaining 80% of staff members. This would increase the probability of getting 3 emergency trained staff after 3 phone calls, as the first call would have a higher chance of reaching an emergency trained staff member.

Ultimately, the best solution would depend on the specific circumstances and the accuracy of the contact list. However, by updating the contact list and prioritizing phone calls to known emergency trained staff members, the probability of getting 3 emergency trained staff after 3 phone calls can be increased.