Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 6 Probability & Statistics
Problem
Solution
1) Trina has to do an environmental art project. She chooses to do one with the bottle caps from gallon milk containers and asks her friends to help collect them. She gets a total of 1000 bottle caps and they come in red - whole milk, blue - 2%, yellow and green - 1%, and white - skim. She mixes them all up in a box and pulls out the first handful of 20 to begin her project. There are 3 red, 5 blue, 3 yellow, 4 green, and the rest are white. If this handful is a good representation of all 1000 bottle caps, about what percent come from 1% milk? 1. Add the yellow and green to get the total 1% caps = 7
2. Divide this by 20 = 7/20 = 35%

Problem
Solution
2) Rina is making a necklace using 3 different colors of beads, Red, White and Blue, and a center stone. How many ways can she string 4 beads using all 3 colors of beads?
Note: Consider that two beads of the same color are interchangeable and don't count as 2 separate combinations 		The wording of this problem is tricky. What it means is that you can duplicate each color to make 4 choices.
Use R for red, W for white and B for blue.
So your 3 choices are:
    2 reds, 1 white, 1 blue (R R W B)
    2 whites, 1 red, 1 blue (W W R B)
    2 blues, 1 red, 1 white (B B R W)
There are 2 ways to approach this problem.

Method 1:List the possibilities for RRWB and then multiply by 3.
List all the possibilities for the first option containing 2 reds. Remember that the two red beads are the same and that a combination of R1 R2 W B is the same as R2 R1 W B.
Two reds:
    R R W B
    R R B W
    R B R W
    R B W R
    R W R B
    R W B R
    W R R B
    W R B R
    W B R R
    B R R W
    B R W R
    B W R R
This is 12 combinations.
Multiply this by 3 (to get the 2 whites and the 2 blues combinations) to get the total number of color combinations =
12 x 3 = 36

Method 2: Use the formula for permutations of words with repeated letters. This is the same problem, except we have beads instead of words.
This formula is :
    P = n! / r! where:
    n = the number of colors (the exclamation point means to take the factorial of n.)
    r = the number of repeated colors (again, factorial)
Once you have this number, multiply by 3 because you can duplicate each of the 3 colors.
    P = 4! / 2! = 24/2 = 12.
    Multiply by 3 to get the total for all 3 colors = 3 x 12 = 36.
The number of combinations is 36

Problem
Solution
3) An 8 by 4 rectangular dart board has a triangle drawn inside of it that has two of its vertices at the midpoints of two sides of the rectangle and the third vertex at a vertex of the rectangle, see picture. If a dart is equally likely to hit any point of the rectangle, what is the probability the dart will land inside the triangle? Express your answer as a reduced fraction. The easiest way to approach this problem is to add up the areas that are not in the shaded area and subtract them from the rectangle area to get the area of the shaded area.
1. Label all sides and half-sides.
2. Compute the area of the white triangles =
    upper left: (4 x 2)/2 = 4
    upper right: (4 x 4)/2 = 8
    lower left: (8 x 2)/2 = 8
    Total: 20 sq. units
3. Subtract this from the area of the rectangle =
    32 - 20 = 12 sq. units.
4. Probability = 12/32 = 6/16 = 3/8
4) Jeron has been increasing his test average by 5 points with every test he takes. The tests are all out of 100 points. His first test score was a 60. He just got back his fifth test. What was his score? You have to compute the scores one by one making the average increase by 5 points with each one:
Solve each one for the appropriate score, then substitute it in the next computation:
(60 + S2)/2 = 65 S2 = 70
(60 + 70 + S3)/3 = 70 S3 = 80
(60 + 70 + 80 + S4)/4 = 75S4 = 90
(60 + 70 + 80 + 90 + S5)/5 = 80S5 = 100
5) To pass the time, Louisa and Maya are playing a game with a 12 sided die numbered 1 through 12 and a 20 sided die numbered 0 to 19. Louisa gets one point if the product of the two numbers is less than 60. Maya gets one point if the product is 60 or higher. What is the probability that Maya will get the point on the next roll? Express your answer as a reduced fraction. The only way to do this is to count the number of products that are 60 or above and divide by the number of possibilities. Even though the table only goes to 19 remember there is a zero column also:
    There are 102 products at or above 60, divided by
    20 x 12 = 240 = 102/240 = 51/120 = 17/40