Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 8 Probability and Statistics
Problem
Solution
1) To inspire good citizenship, the teacher places popsicle sticks with the student names on them in the citizenship cup when they display acts of good citizenship. At the end of the day five names are drawn at random for prizes. On Friday there were 20 popsicle sticks in the citizenship jar including Miguel. What is the probability that Miguel will be picked first or second for a prize?
  1. P(a) or P(b) = P(a) + P(b)
  2. Miguel's probabilities are:
        1⁄20 +     1⁄19 = 39⁄380 if the teacher did not put the first popsicle stick back.
  3. Miguel's probabilities are:
        1⁄20 +     1⁄20 = 2⁄20 if the teacher does put the first popsicle stick back.
2) Douglas MS participated in a contest with the results shown. If five teams participated, what is their highest score? (Try and create a data set that works.)
  1. OK, we have a data set of 5 numbers whose median is 78, whose mean is 75 and whose range is 8.
  2. Since there are 5 scores and the median is 78 then the middle score is 78
  3. The mean of the other 4 scores is (75x5 - 78) / 4 =
        (375 - 78) / 4 = 297/4 = 74.25
  4. This mean is lower than the median, so the set looks like:
        70 X 78 78 78:
  5. Using the mean of 75 and solving for X:
        X = (75x5) - (3x78 + 70) = 71
  6. This works! The set is:
        70 71 78 78 78
    So the maximum score is 78

3) A Elite Style inspects 500 hair dryers manufactured and finds 495 to have no defects. There are 20,000 hair dryers in their warehouse. Predict the number of hair dryers that will have no defects in the warehouse. 		(495 / 500) x 20,000 = 19,800

Problem
Solution
4) When rolling a single six sided die three times, how much more likely are you to roll all three different numbers compared to all three the same number? Express your answer as a ratio.
  1. Probability of rolling 3 different numbers:
        5⁄6 x 4⁄6= 20⁄36 = 5⁄8
  2. Probability of rolling all 3 the same number:
        1⁄6 x 1⁄6 + 1⁄36 =
  3. The ratio of these probabilities is:
        5⁄8 ÷ 1⁄36 = 5⁄8 x 36⁄1 = 5 x 4 = 20 = 20:1
5) Dana has to collect experimental data on 3 penny flips. Based on her data she is a little suspicious of her penny. What should the data look like in theory for a fair penny in 40 trials?
  1. With 3 flips, the possibilities are:
        H   H   H
        H   H   T
        H   T   H
        H   T   T
        T   T   T
        T   T   H
        T   H   T
        T   H   H
  2. Probability of 3 heads in 40 trials = 1⁄8 x 40 = 5
  3. The probability of 2 heads and 1 tail = 3⁄8 x 40 = 15
  4. The probability of 2 tails and 1 head = 3⁄8 x 40 = 15
  5. Probability of 3 tails in 40 trials = 1⁄8 x 40 = 5