Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 7 Probability and Statistics
Problem
Solution
1) Jen's warehouse receives a shipment of 700 refurbished toasters. She tests 40 of them at random for defects and finds 2 with power supply failure. If this sample is representative of the entire shipment, how many toasters should Jen expect to have power source failure? (2/40)(700) = 700/20 = 35

2) A set of five 2-digit whole numbers has a mean of 16, a mode of 13, and a median greater than the mean. What is the largest possible value of the largest number of the set?
  1. The sum of all 5 2-digit numbers is 5x16 = 80
  2. There must be at least 2 13s, so the other 3 must add to 80 - 26 = 54 and their average must be 54/3 = 18
  3. For the median to be greater than the mean (16) it must be the lowest value possible to maximize the size of the last (largest) number. So, it must be 17.
  4. The next 2 numbers must be higher than 17 to make 17 the median (middle number)
  5. So far our set looks like this:
        13 13 17 X Y
  6. The last 2 numbers (X and Y, the one we are trying to maximize) must total 80 - 43 = 37
  7. Notice that if Y is 20, then X must be 17, which can't be, because we already have a 17 and that would make it the mode also.
  8. Y also can't be greater than 20 or X must be less than 17, making it the median.
  9. Therefore Y must be 19 to fit all the requirements.
    Here is the set: 13 13 17 18 19

Problem
Solution

3) You have a bag containing 5 red candies, 7 blue candies, 8 green candies, and 12 yellow candies. If you pull candies from the bag at random, what is the minimum number you would have to take to ensure getting 3 of at least one color?
  1. To insure getting 3 of at least one color, you have to take 2 sets of 1 each of all 4 colors = 8 candies.
  2. This results in 2 of each color
  3. The next candy is guaranteed to match one of the 4 colors, so the minimum number is 8+1 = 9
4) A six-sided die is weighted so it rolls 6 five times as often as any other number. If it rolls 6 five times in a row, what is the probability it will roll 6 on the sixth roll?
  1. The fact that it has rolled 6 5 times in a row is irrelevant. That is in the past.
  2. There are 5 other numbers and the 6 rolls 5 times any other number, so the possible rolls are:
        1,2,3,4,5,6,6,6,6,6, making the probability that it will roll a 6 (5/10) = 50%

5) There are 5 little league baseball teams in your county. There are 4 teams in a neighboring county. Each team will play three games against each of the other teams within their county and two games against each of the teams in the neighboring county. How many total games will be played by the end of the season? 		Note: When selecting 2 teams out of a league to play each other, that is a combination.
When selecting 2 teams that are not in the same league, that is a permutation
  1. Games in your county:
        Each team plays the other team 3 times: ((5x4)/2)x 3 = 30 games
  2. Games with neighboring county teams:
        Each team plays the other team twice: (5x4)x2 = 40 games
  3. Games the neighboring county plays with themselves = (4x3)/2 = 6 x 3 games = 18 games total
  4. The total number of games played is 30 + 40 + 18 = 88 games