May 20, 2018

Event A: $\frac{5}{6}$
Event B: $\frac{11}{18}$

#### Explanation:

Event A: Their sum is greater than 7

We can solve through complementary events.
P(sum greater than 7) = 1- P(sum equal to 7)

What are the numbers that equal to 7?

1,6
2,5
3,4
4,3
5,2
6,1

In each case, you have $\frac{1}{36}$ chance of rolling those two numbers. Here's why:

Looking at 1,6

You have $\frac{1}{6}$ chance of rolling a 1 and you also have $\frac{1}{6}$ chance of rolling a 6. Since these two chances are independent events, then you multiply the numbers together. $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

The $\frac{1}{36}$ applies to all 6 numbers. So $\frac{1}{36} \times 6 = \frac{1}{6}$

Therefore, P(sum equal to 7) = $\frac{1}{6}$

So, P(sum greater than 7) = $1 - \frac{1}{6} = \frac{5}{6}$

Event B: the sum is not divisible by 4 and not divisible by 6

The largest sum you can have between two numbers is 12 ie 6+6 and the smallest number you can have is 2 ie 1+1

So numbers divisible by 4 in that range are: 4, 8, 12
and numbers divisible by 6 in that range are: 6, 12

Therefore, looking at the table below, you can see that there are 14 combinations that you cannot throw. Since each combination ie 1,3 or 6,6 and you have $\frac{1}{36}$ chance of throwing each combination, then P(sum is divisible by 4 and 6) is equal to$\frac{1}{36} \times 14 = \frac{7}{18}$

P(sum not divisible by 4 and 6) = 1-P(sum is divisible by 4 and 6)
P(sum not divisible by 4 and 6) = $1 - \frac{7}{18} = \frac{11}{18}$

May 20, 2018

$P \left(A\right) = \frac{5}{12}$

$P \left(B\right) = \frac{11}{18}$

#### Explanation:

Consider all the possible results of two dice rolls.

$1 + 1 = 2$
$1 + 2 = 3$
$\cdots$
$2 + 1 = 3$
$2 + 2 = 4$
$\cdots$
$6 + 5 = 11$
$6 + 6 = 12$

In total, there are $36$ combinations:

$2 , 3 , 3 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 6 , 6 , 6 , 6 , 6 , 7 , 7 , 7 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 9 , 9 , 9 , 9 , 10 , 10 , 10 , 11 , 11 , 12$

$15$ out of these $36$ are strictly greater than 7.
$22$ out of these $36$ are not divisible by 4 or by 6.

Therefore,

$P \left(A\right) = \frac{15}{36} = \frac{5}{12}$

$P \left(B\right) = \frac{22}{36} = \frac{11}{18}$