Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 7 Probability and Statistics
Problem
Solution
1) A set of 5 whole numbers all more than 0 but less than 10 has a mean of 5, a mode of 2, and a median less than the mean. What is the set of numbers?
  1. If the mode is 2 then the other 3 numbers must add to (25-4)/3 = 7
  2. If the median is less than the mean (5) then the median must be 3 or 4
  3. If the median is 3, then the last 2 numbers must add to (25-7) = 18, so the last 2 numbers can be 9 and 9, making the set:
        2   2   3   9   9
    But this cannot be, because 9 would also be the mode!
  4. If the median is 4, then the last 2 numbers must add to (25-8) = 17, so the last 2 numbers can be 8 and 9, making the set:     2   2   4   8   9
2) Greg tosses 4 pennies in the air and they come up 3 heads and 1 tail. Express the probability as a reduced fraction.

 The best way of solving a problem like this is just listing the possibilities.
Using H for heads and T for tails, here is the table of possibilities.
Those with 3 heads and 1 tail are marked (*):
    H   H   H   H
    H   H   H   T *
    H   H   T   H *
    H   H   T   T
    H   T   H   H *
    H   T   H   T
    H   T   T   H
    H   T   H   T
    T   H   H   H *
    T   H   H   T
    T   H   T   H
    T   H   T   T
    T   T   H   H
    T   T   H   T
    T   T   T   H
    T   T   T   T     The probability is 4/16 = 1⁄4

Problem
Solution
3) In a video game, each fireball launched has a 75% chance of hitting a target. What is the probability that exactly 2 fireballs will hit the target when 2 fireballs are launched? Express your answer as a reduced fraction or as a decimal to the hundredths place.
  1. .75 = 3⁄4
  2. The probability is
        3⁄4 x 3⁄4 = 9⁄16 or 0.56
4) A group of 36 students are interviewed about some items they would not want to do without. These were the results rounded to the nearest whole number percent. What is the smallest number of students that could have selected toilet paper and zippers as items they don't want to do without?
  1. Toilet paper: 69% of 36 = 25 students
  2. Zippers: 42% of 36 = 15 students
  3. 25 + 15 = 40 so there must be SOME overlap in order to keep the total number of students 36.
    That overlap is 40 - 36 = 4 students
Notes:
  1. If the number of students that cannot do without toilet paper and the number that cannot do without zippers had added to LESS than 36, then there would be no necessary overlap.
  2. Also note that the data about pizza and tape was irrelevant.
    This is often the case with data analysis. You must discern what information is relevant (toilet paper and zippers) and which is not (pizza and tape).
  3. Also note that it is entirely possible that ALL the students that couldn't do without zippers also couldn't do without toilet paper because their percentage is less than the toilet paper percentage.
5) A store in Vancouver, Washington finds to its disgust that they have managed to collect many Canadian pennies. They decide to hand them back to the public since that's where they came from in the first place. To make it less obvious they divide them up between tills so that the mix of Canadian to American pennies is about 1 to 3. You get a small handful of change from this store consisting of 2 quarters, 1 nickel, 3 dimes and 8 pennies. If you let your little brother randomly pick one coin from this handful, what's the chance he ends up with a Canadian penny?
  1. The probability of getting a Canadian penny in change is 1⁄3
  2. 8 x 1⁄3 = 8⁄3 = 2 pennies
  3. There are a total of 14 coins (your little brother can easily pick out a quarter) so the the probability is:
        2⁄14 = 1⁄7