Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Probability and Statistics
Problem
Solution
1) Billy and Bobbi are playing a dice game. Billy rolls a six sided die four times and adds the face values together. Bobbi rolls a 12 sided die twice and adds the face values together. What are the odds of Billy rolling a 23 compared to Bobbi rolling a 23?
  1. There are 4 ways out of 1296 possibilities with 4 dice to roll a sum of 23:
    6 6 6 5
    6 6 5 6
    6 5 6 6
    5 6 6 6
  2. There are 2 ways out of 144 with 2 12-sided dice to roll a sum of 23:
    12 11
    11 12
  3. Billy's probability is 4⁄1296 = 1⁄324
  4. Bobbi's probability is 2⁄144 = 1⁄72
  5. Take the ratio of the 2 probabilities:
    1⁄324 ÷ 1⁄72 = 72⁄324 = 2⁄9 or 2:9
Note: The problem is really asking for the probability, not the odds. Odds are the number of things you are looking for over the number of things that you are not looking for, where probability is the number of things (or possibilities) that you are looking for divided by the total number of possibilities. These are 2 different concepts.
2) Jordi's math class of 20 students took a quiz. The average score was 80 points out of 100 points. The range of the quiz scores separating the low score from the high score was 25 points. Jordi wonders what the lowest score could be. What is the lowest possible score?
  1. The lowest possible score is achieved when the other 19 scores are maximized.
  2. Calling the lowest score L, the maximum score is L+25
  3. Assuming all the other scores are the maximum, then the average is:
    (L + 19(L + 25)) / 20 = 80
    20L = 1600 - 475 = 1125
    L = 56.25, rounded up to 57

Problem
Solution
3) There are ten jelly beans in a sack. Four are mint flavored, three are celery flavored, two are licorice flavored, and the last is peanut butter flavored. Nyang, who doesn't like peanuts, but loves celery, reaches in and grabs two jelly beans and eats them without looking. What is the probability that one was celery flavored, and the other was not peanut flavored?
  1. Method 1: Brute force! The jellybeans are (each one is numbered) where
    C=celery, M=mint, L=licorice and P=peanut:
    C1 C2 C3 M1 M2 M3 M4 L1 L2 P
    Here are the possibilities:
    (the ones that have a celery (Cn) and not a peanut (P) are marked with an *)
    C1 C2*
    C1 C3*
    C1 M1*
    C1 M2*
    C1 M3*
    C1 M4*
    C1 L1*
    C1 L2*
    C1 P
    		 C2 C3*
    C2 M1*
    C2 M2*
    C2 M3*
    C2 M4*
    C2 L1*
    C2 L2*
    C2 P
    		 C3 M1*
    C3 M2*
    C3 M3*
    C3 M4*
    C3 L1*
    C3 L2*
    C3 P
    		 M1 M2
    M1 M3
    M1 M4
    M1 L1
    M1 L2
    M1 P
    		 M2 M3
    M2 M4
    M2 L1
    M2 L2
    M2 P
    		 M3 M4
    M3 L1
    M3 L2
    M3 P
    		 M4 L1
    M4 L2
    M4 P
    		 L1 L22
    L1 P
    		 L2 P
    There are a total of 45 possibilities of which 8+7+6 = 21 of them have a celery and not a peanut for a probability of 21/45 = 7/15
  2. Method 2: Count the combinations:
    1. The total number of combinations of 10 things taken 2 at a time is
      (10 x 9)/2 = 45
    2. With each of the celery jellybeans, there are 8 + 7 + 6 = 21 possible combinations that do not include a peanut flavored bean.
    3. Therefore the probability is 21/45 = 7/15

Problem
Solution
4) Many Flavors Gelato Bar has 7 flavors of gelato today, 2 kinds of gooey sundae sauces, and a choice of chocolate sprinkles or chopped nuts topping. Across the street Mama's Ice Cream Parlor has 6 flavors of ice cream, 3 kinds of gooey sundae sauces, and a choice of two kinds of sprinkles for topping, chocolate or fruity. Selena likes having lots of choices. If she is going to make a sundae out of 2 scoops of gelato or ice cream, one sundae sauce and one topping, which store will give her more options and how many more? She can choose to have both scoops be the same flavor.
  1. Many Flavors: 7x7x2x2 = 196
  2. Mama's: 6x6x3x2 = 216
  3. Mama's gives her (216 - 196) = 20 more choices.

5) Sweet Raisins sells 1.5oz boxes of raisins. Not all the boxes have the same number of raisins. In a quality control check 25 boxes were randomly selected and checked for weight and number of raisins. The graph displays the data for the number of raisins. Compare the distance between the furthest point out and the mean to the mean. Express your answer as a percent to the nearest whole number.
  1. The data set is:
    16 16 16 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 19 20 20 21 22 22 25
  2. The mean is
    16x3 + 17x8 + 18x7 + 19 + 20x2 + 21 + 22x2 + 25 = 459/25 = 18.36
  3. The furthest point out is 25
  4. (25 - 18.35)/18.36 = 0.3616 = 36%