- #1

erisedk

- 374

- 7

## Homework Statement

For the three events A, B and C, P (exactly one of the events A or B occurs) = P (exactly one of the events B or C occurs) = P (exactly one of the events C or A occurs) = p ,

and P (all the three events occur simultaneously) = p

^{2},

where 0 < p < ½ .

Then the probability of at least one of the three events A, B and C occurring is:

Ans: ## \dfrac{3p+2p^2}{2} ##

## Homework Equations

## The Attempt at a Solution

http://s3.amazonaws.com/minglebox-photo/core-0000-c88370190d4b414d010d4b415d220010.data-0000-fdbffe7622c53ecd0122c5c50d0b0334.gif

NOTE: The regions shown do not overlap with each other, i.e. P(A) ≠ region 1, instead P(A) = region ( 1 + 2 + 4 + 5).

By symmetry, I assume region 1 = 3 = 7 = p/2

Region 5 = p

^{2}

Again by symmetry, region 2 = 4 = 6 = x.

I need to find x because the probability that I got to calculate is the sum of all the regions, ie.

## \dfrac{3p}{2} + 3x + p^2 ##

I don't know how. I initially though that the sum of all probabilities might be 1, but then it doesn't say that this is the case. For all we know, there might be 10 more events. So, I dismissed that option. I don't know how to proceed from here.