occurence of Independent Events

[Bob12]

There are 3 people on a team who are experiencing independent events with 4 types of event outcomes:

“Big reward”, “Little Reward”, “Break Even”, and “Loss”.

Every person has a specific event outcome probability which is known to us:

Person #1 :

Break Even: 45%
Big Reward: 25%
Loss: 15%
Little Reward: 15%

Person #2 :

Break Even: 50%
Big Reward: 5%
Loss: 20%
Little Reward: 25%

Person #3 :

Break Even: 20%
Big Reward: 30%
Loss: 15%
Little Reward: 35%

I am trying to get an understanding of the likelihood of an event occurring 1 or more times among the group.

So in my mind if you said “What is the likelihood that any of the three people will experience a big reward outcome for their event?”

that would look like this:
0.05 + 0.25 + 0.3 - (0.05 * 0.25) - (0.05 * 0.3) - (0.25 * 0.3) + (0.05*0.25*0.3) = 0.50125

Applying that thinking across all event outcomes you end up with :

Big Reward: 50.125%
Little Reward: 58.56%
Break Even: 58%
Loss: 42.2%


That doesn't seem correct. I was hoping someone could explain what I am missing

thanks for your time!

 

[Dr.Peterson]

I am trying to get an understanding of the likelihood of an event occurring 1 or more times among the group.

So in my mind if you said “What is the likelihood that any of the three people will experience a big reward outcome for their event?”

that would look like this:
0.05 + 0.25 + 0.3 - (0.05 * 0.25) - (0.05 * 0.3) - (0.25 * 0.3) + (0.05*0.25*0.3) = 0.50125

I believe you are using the inclusion-exclusion principle here; it would be easier to follow if you used the same order of events as in the problem, and said in words what you are doing.

It's easier to use complements: The probability of at least one person getting a big reward is 1 minus the probability that none of them get a big reward: 1 - (1-0.25)(1-0.05)(1-0.30) = 1 - (0.75)(0.95)(0.70) = 1 - 0.49875 = 0.50125

But you got the right answer.

Your problem probably lies in having a wrong expectation of how the various probabilities interact. The events are not mutually exclusive: It is possible for at least one to get the big reward, and at least one to get the little reward, and so on.

 

[Bob12]

Thank you for the help, Dr. Peterson!

I knew they were not mutually exclusive but I am very surprised that despite the people having very a distinct weighting to a given event, that as a group they would all smear into "anything is possible".

if I changed it to a mutually exclusive problem, everything becomes much more distinctive. I had expected something closer to that so I thought I was very wrong.

 

[Dr.Peterson]

Probability is a field in which your intuition is probably wrong ...

I'm always being surprised by results. And I'm supposed to know better. (Here, though, I wasn't surprised at all.)