Probability...~ (help check..)?

Question

1) A basket contains 10 apples and of these, two are rotten at the core. An apple is picked at random. What is the probability that it is rotten?

My answer is 2/10 = 1/5.

2) Two ordinary dice are thrown. Find the probability that:

(a) the sum on the two dice is 3;

My solutions:

2/36 = 1/18

(b) the sum on the two dice exceeds 9;

[No idea.]

(c) the two dice show the same number;

My solutions:

6/36 = 1/6

(d) the numbers on the two dice differ by more than 2;

[No idea.]

(e) the product of the two numbers is even.

[No idea.]

Those questions that I have wrote 'No idea' is because I found myself writing down every single conditions that it will happen (eg. (1,1), (2,1) ) I wrote down them ONE BY ONE which is a complete waste of time. In exam, I do not have so much time to do these. So, anyone has any short-cuts (easy methods) that I can adhere to so I would not find probability to be a hard subject?

Thanks. Please explain too.

Answer 1

1. 1/5

2.

a. (1,2) or (2,1)

2/36 = 1/18

b. It must be 10, 11 or 12

You can get 10 in 3 ways (6,4), (5,5), (4,6)

You can get 11 in 2 ways (5,6), (6,5)

You can get 12 in 1 way (6,6)

(3+2+1)/36 = 1/6

You need to list all conditions for these to be met. In questions like these it will rarely exceed 6 or 7 possibilities that you should be able to list in a logical order to make sure that you don't miss any possibilities.

c. Your answer is correct

d. There might be a short cut for this but the quickest way would be to list all the possibilities again. There are just 6 pairs. Double that because the order matters.

The probability is 2×6/36 = 1/3

e. odd×odd = odd

even×odd = even

even×odd = even

Find the probability of getting an odd number

(3/6)×(3/6) = 1/4

The probability of getting an even number is 1 - 1/4 = 3/4