S1 (Edexcel) Probability

it's given that P(C) = x/(x+5) and P(D)= 5/x and that x is a positive constant.

now how do i show considering P(C) + P(D) that they cannot be mutually exclusive?

Show their sum is greater than 1.

can you explain how do you know that their sum will be greater than 1?

because in the previous part they provided a venn diagram for different events:


and when i add P(A) + P(B) = 7/25 + 1/5
i get 0.48 which is less than 1

it's given that P(C) = x/(x+5) and P(D)= 5/x and that x is a positive constant.

now how do i show considering P(C) + P(D) that they cannot be mutually exclusive?

P(C)+P(D) ...

P(C)+P(D) = (x^2 + 5x + 25)/(x^2 +5x)
= 1 + 25/(x^2 +5x)

assuming C and D are mutually exclusive the sums of their probabilities is greater than 1 since x>0 thus they cannot be mutually exclusive

C and D are different. Mutual exclusive is tied to
p(CuD) = p(C)+p(D)-p(CnD)
so if theyre mutually exclusive their intersection is zero and therefore their sum must be <= 1 as were dealing with probabilities so 0<=p<1.

So show
5/x + x/(x+5) > 1
obviously the x must be a certain interval for them to be probabiities in the first place, then the usual ways of going about it are
* combine and reason about the expression
* note the left hand side is nearly a partial fraction expression, except the second term isnt proper - does it help if you make it proper. Number spotting, the repeated 5 looks suspicous.

Post what you try/do.

i tried to understand the explanations, and i did understand the first part of what you both were trying to convey.

BUT
however i didn't quite still understand why would their summation be greater 1 and what does that have to with x being a positive number.

i tried working out this problem again with whatever i understood so far, but i'm just not sure if this is a right way.

my working:
if C and D are mutually exlusive then their intersection would be 0. so that would mean then: P(CUD) = P(C) + P(D)
and their sum of probability has to be greater than 0 and less than or equal to 1

so this is what i did:
(x^2 + 5x + 25)/(x^2 +5x) <= 1
and at the end i get:
25<=0
this cannot true, so therefore i come to the conclusion that they cannot be mutually exclusive.
:/

Its probably slightly easier to do the 2nd approach and argue that x>=5 for p(C) and p(D) to be probabilities. Then
p(C) + p(D) = 5/x + x/(x+5) = 5/x + (x+5-5)/(x+5) = 1 + 5/x - 5/(x+5) > 1
as (5 <) x < x+5. The 5/(x+5) is just a shifted (translated) 5/x reciprocal so the inequality is simple. So they cant be mutually exclusive.

just one more thing...
just to be clear, based on the question, we're considering outside the circles as zero, right?
like P(C' intersection D') = 0