What portion of the socks sold were purple

Pairs of socks sold
Black:9
Blue:20
Gold:6
Green:22
Purple:25
Red:18

What portion of the socks were blue?

If Aliza wants to order 1200 pairs of socks, how many red socks should she order?

Player A and player B decide to use the following rule when rolling two fair dice: player A wins when any number is rolled that is divisible by 2. Player B wins when any number is rolled that is divisible by 3. Is this a fair game? If not, who has the unfair advantage?

A. No, Player B has an unfair advantage because 12 is divisible by 3, 4 times.
B. No, No one has the advantage, but no one can win if they roll a 1, 5, 7, or 11.
C. No, Player A has an unfair advantage because 12 is divisible by 2, 6 times.
D. Yes, it is a fair game.

Hey, b00ty,

PART A: SOCKS

In all, there were 9 + 20 + 6 + 22 + 25 + 18 = 100 pairs of socks sold. Since 20 of these were blue, the proportion of blue socks to total socks sold was 20/100 = 1/5.

The proportion of red socks sold was 18 pairs of red socks out of 100 pairs, or 18/100. If Aliza wants to order 1,200 pairs of socks, she will want x red socks out of 1,200, a proportion of x/1200.

Since she wants to keep the proportions equal, then:

18/100 = x/1200

Solving for x,

x = (1200) (18/100)
= 12 × 18
= 216 pairs of red socks.


PART B: DICE

There are 36 possible outcomes of the two-dice roll:
{[1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6],
[2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [2, 6],
[3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6],
[4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6],
[5, 1], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6],
[6, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6]}

Now, 6 of these outcomes have sums divisible by both 2 and 3, resulting in a tie:

{[1, 5], [2, 4], [3, 3], [4, 2], [5, 1], [6, 6]}

There are 18 outcomes whose sums are divisible by 2. Removing the 6 outcomes above that are also divisible by 3, we are left with 12 outcomes resulting in a win for Player A:

{[1, 1], [1, 3], [2, 2], [2, 6], [3, 1], [3, 5],
[4, 4], [4, 6], [5, 3], [5, 5], [6, 2], [6, 4]}

Since Player A has 12 winning outcomes out of 36 possible outcomes, the probability that Player A will win on a particular roll is 12/36 = 1/3.

There are only 12 outcomes whose sums are divisible by 3; however, as before, 6 of these are also divisible by 2. Removing these, we are left with 6 outcomes that result in a win for Player B:

{[1, 2], [2, 1], [3, 6], [4, 5], [5, 4], [6, 3]}

So, the probability that Player B will win on a particular roll is 6/36 = 1/6.

Since Player A wins 1/3 of the time and Player B wins only 1/6 of the time, Player A wins twice as often as Player B, and has a clear advantage.

So, it is not a fair game, and the best answer is C.