math - Probability of a sample space -

i have been doing exam review class questions database no solutions given. i'm kind of confused answer be

> consider sample space s={a,b,c,d} , probability function pr: s->|r on s. events: a={a}, b={a,b}, c={a,b,c}, d={b,d}

you given pr(a)=1/10, pr(b)=1/2, , pr(c)=7/10 pr(d)? show work.

i thought pr(d)=1-(pr(a) + pr(b) + pr(c)) 3 probabilities equal more 1. tried looking @ superset of s still couldn't find answer.

what answer , why?

the answer p(d) = 7/10

to find this, @ how samples combined in brackets create events, , note a+b+c+d=1, lowercase letters not uppercase ones. each of given events find

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

p(d) = 4/10 + 3/10 = 7/10

And