"either but not both" ??

rlm

New Member
#1
Hi,
I was trying to complete this question, but I'm confused about where either but not both comes in:

"certain chickens on a large farm have developed diseases, 2% have fatty liver syndrome, 3% have cage layer fatigue. 4% of the chickens have either one of these conditions, but not both"
a) Find the percentage of poultry with both conditions
b) a chicken is selected at random from this farm, what is the probability it has neither condition?

Calling 'fatty liver syndrome' event A, and 'cage layer fatigue' event B
What I have understood so far is that:
P(A)=0.02
P(B)=0.03

P(A) + P(B) = 0.04

Where this fits in a Venn diagram, I'm not too sure- since they are not mutually exclusive, there is P(AnB), but using the addition rule requires P(AuB), the conditions of which mean that A and B occur together, which is not equal to 0.04?

thanks!
 

Dason

Ambassador to the humans
#2
Hi,

What I have understood so far is that:
P(A)=0.02
P(B)=0.03
These look fine

P(A) + P(B) = 0.04
This is wrong. Clearly this can't be true in light of what what we have at the top (which adds up to .05). The condition related to .04 is "have either one of the conditions - but not both" which means the chicken could have A but they don't have B, OR they could have B but they don't have A.
 

rlm

New Member
#3
Thanks!
So if A' is the complement of A and B' is the complement of B,
then the 0.04 is equal to P(AnB')+ P(BnA'),
So can that, as well as P(A) which includes all A, and P(B) which is all B, be used to find the P(AnB)?

P(AnB) = P(A) + P(B) - [P(AnB')+ P(BnA')]

on a Venn diagram, how would this work? P(A) would be the whole circle, and P(B) would be another whole circle interlapping with A and together the P(A) + P(B) would include P(AnB)?

thanks!