## Monday, December 24, 2012

### 2013 Mathcounts State Prep: Counting and Probability

Question #1: Rolling two dice, what is the probability that the product is a multiple of 3?
Solution I :
As long as one of the two numbers turn up as 3 or multiple of 3 (in this case, a "6"), the product of the two numbers will be a multiple of 3.
There are 6 * 6 = 36 ways to get 2 numbers. Out of the 36 pairs, you can have
3 - 1
3 - 2
3 - 3
3 - 4
3 - 5
3 - 6, 6 ways.
However, there are only 5 ways left if the other die has a 3 since 3 - 3 only counts as once.
1 - 3
2 - 3
4 - 3
5 - 3
6 - 3
Next we look at "at least one number is "6".
6 - 1
6 - 2
6 - 4 (We already used 6 - 3)
6 - 5
6 - 6 so 5 ways.
The other way around, we have
1 - 6
2 - 6
4 - 6
5 - 6  Total 4 ways, so the answer is $$\frac{\Large{( 6 + 5 + 5 + 4 )}}{\Large{36}}$$ = $$\frac{\Large{ 20}}{\Large{36}}$$ = $$\frac{\Large{5}}{\Large{9}}$$.

Solution II:
The easiest way to solve this problem is to use complementary counting, which is 1 (100% or total possible way) - none of the the multiples of 3 showing up, so 1 - $$\frac{\Large{4}}{\Large{6}}$$ *$$\frac{\Large{4}}{\Large{6}}$$ = 1 - $$\frac{\Large{2}}{\Large{3}}$$ * $$\frac{\Large{2}}{\Large{3}}$$=  $$\frac{\Large{5}}{\Large{9}}$$

Question #2: [2002 AMC-12B #16] Juan rolls a fair regular eight-sided die. Then Amal rolls a fair regular six-sided die. What isthe probability that the product of the two rolls is a multiple of 3?
Solution:
Using complementary counting (see solution II of the previous question), 1 - $$\frac{\Large{6}}{\Large{8}}$$ *$$\frac{\Large{4}}{\Large{6}}$$ = 1 - $$\frac{\Large{3}}{\Large{4}}$$ * $$\frac{\Large{2}}{\Large{3}}$$=  $$\frac{\Large{1}}{\Large{2}}$$