Probability of a sample space [closed]

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I have been doing an exam review for my class where the questions are from a database but no solutions are given. I'm kind of confused as to what the answer to this would be

> Consider a sample space S={a,b,c,d} and a probability function Pr: S->|R on S. Events: A={a}, B={a,b}, C={a,b,c}, D={b,d}

You are given Pr(A)=1/10, Pr(B)=1/2, and Pr(C)=7/10 What is Pr(D)? Show your work.

I thought Pr(D)=1-(Pr(A) + Pr(B) + Pr(C)) but those three probabilities equal to more than 1. I tried looking at the superset of S but I still couldn't find the answer.

What is the answer and why?

Answer 1

The answer is P(D) = 7/10

To find this, look at how the samples are combined in the brackets to create events, and note that a+b+c+d=1, the lowercase letters not the uppercase ones. So from each of the given events we find

1/10 = a
5/10 = 1/10 + b => b = 4/10
7/10 = 1/10 + 4/10 + c =>  c = 2/10
P(D) = 4/10 + d

Since

a + b + c + d = 1
d = 1 - a + b + c = 1 - (1/10 + 4/10 + 2/10) = 1 - 7/10 = 3/10

So

P(D) = 4/10 + 3/10 = 7/10