Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 7 Probability and Statistics
Problem
Solution
1) Miguel entered a contest offering prizes to the top 3 finishers. The probability of winning 1st place is 12%, the probability of winning 2nd place is 18%, and the probability of winning 3rd place is 20%. What is the probability that Miguel will not win any prize? Express your answer to the nearest percent. 12% + 18% + 20% = 50%

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) An airplane is built to be able to fly on one engine. If the plane's two engines operate independently, and each has a 1% chance of failing in any flight, what is the chance a plane would fail to complete a flight? Express your answer as a percent. You can't do arithmetic with percentages until you convert them into either fractions or decimals. We'll use fractions. Convert the probabilities to fractions and multiply the 2 probabilities together:
    (1/100) x (1/100) = 1/10000 = 1/100 % = .01%

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. All 3 are the same number:
        Probability of the second roll matching the first: 1⁄6
        Probability of the third roll matching the first: 1⁄6
        total = 1⁄6 x 1⁄6 = 1⁄36
  2. All are different.
    The first roll establishes one number,
    the second has 5 remaining numbers and
    the 3rd 4 remaining numbers:
        5⁄6 x 4⁄6 = 20⁄36
  3. Divide (1/36) by 20/6
        1⁄6 x 6⁄20 = 1⁄20 = 1:20

5) There are 25 students in Mr. B.'s class. Tori is one of the students who has a $2.00 weekly allowance. Which average, mean or median, should she report to her parents to encourage them to give her a higher allowance? Write your choice and its value.
  1. Compute the mean:
        0x3 2x5 + 2.25x2 + 2.5x4 + 3.5x4 + 4x3 + 3.75x2 + 5x1 + 7.5x1 = 70.5/25 = $2.82
  2. Compute the median. This is the 25/2 + 1 = 13th allowance, which is (counting from the left) $2.50.
  3. Tori should go for the average (the mean) = $2.82