probability part 6-_-?

1.
Probabilities
student Geography History
May 2/3 1/2
Kate 5/12 3/4
(a) Both fail in both subjects.
use multiplication rule
(1-2/3)(1-1/2)(1-5/12)(1-3/4)
=(1/3)(1/2)(7/12)(1/4)
=7/288
(b) At least one does not fail in both
=> not both of them fail in both
So Probability
=1-7/288
= 281/288

2. Use a tree diagram to show that
M=MTR
L=late
Contingency table:
Option MTR(M) Bus (~M)
Late (3/5)(1/4) (1-3/5)(2/5)
Not late (3/5)(3/4) (1-3/5)(3/5)
P(L)=P(M and L) + P(~M and L)
=(3/5)(1/4) +(1-3/5)(2/5)
=3/20+4/25
=31/100

3. Need to know how many spaces are there, or is it "infinite".

4.
There are 12!=479001600 ways to arrange 12 persons.
If the girls are not to be next to each other, we place the 8 boys first (in 8! ways), leaving 9 positions for the first girl to be placed, 8 for the next, then 7, then 6 for a total of 9!/5!.
So the number of arrangements for the girls to be separated is 8!9!/5!=121927680
So probability that girls are separated
=121927680/479001600
=14/55