Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 7 Probability and Statistics
Problem
Solution
1) Mac is a Harry Potter fan and buys a box of Bertie Bott's Every Flavored Beans. He shakes out a handful. 6 are bogey flavored, 12 grass flavored, 2 chocolate, 9 earwax and 4 sardine flavored. He eats 3 of them. What is the probability that the 3rd one he eats is chocolate, assuming the first two were not chocolate?
  1. His handful was 6+12+2+9+4 = 33 beans
  2. He eats 2 of them, so there are 31 left
  3. There are still 2 chocolate beans left out of 31 so the probability is 2⁄31

2) Mr. Domani's class is playing multiplication bingo. Two dice are rolled and the numbers on the dice are multiplied together to get a product. Anna's board is pictured with the numbers she has crossed as products that have been rolled. What is the probability she will have a bingo with the next roll of the dice?
(Bingo means a row, column, or diagonal are all crossed out.)
  1. there are 2 numbers that will produce a bingo:
    1. 15 making a vertical bingo on the left.
      There are 2 ways to roll a 15: 3x5 and 5x3
    2. 12 making a horizontal bingo on the bottom.
      There are 4 ways to roll a 12: 4x3 3x4 6x2 2x6
  2. This is 6 ways to get a bingo out of 36 (6x6) possible rolls
  3. The probability is 6/36 = 1⁄6
3) All 8th graders take a test and your teacher shows your class the results in a histogram and gives the exercise: represent the same data with a pie chart. Tina notices that one of the sections has an angle close to 90 degrees. What is the range of scores that goes with that section?
  1. An angle of 90 degrees in the pie chart corresponds to 25% of the scores.
  2. There were 20 + 70 + 90 + 58 = 238 students.
  3. 25% of 238 is 59.5 so the 58 students that scored 76-100 were 25% of the students.

Problem
Solution
4) Tina's teacher gives a 20 point quiz every Friday. On her first five quizzes, Tina has scores of 15, 4, 16, 18, and 15. After the seventh quiz, her mean score is 15. What is the median?
  1. Re-arrange the scores in numeric order:
        4 15 15 16 18
  2. There are 2 more scores to add. Suppose one of them is a 15. Then the scores would be:
        4 15 15 15 16 18 x
  3. The equation for the mean is:
        (4 + 15 + 15 + 15 + 16 + 18 + x) / 7 = 15, so
        (83 + x) / 7 = 15
        83 + x = 105
        x = 22, which is impossible because the most points available are 20, so the numbers (both of them) must be above 16 , like this:
        (4 + 15 + 15 + 16 + 17 + 18 + x) / 7 = 15, so
        (85 + x) / 7 = 15
        x = 105 - 85 = 20, the highest possible score.
  4. Since both numbers added must be greater than 16, that makes 16 the median.
5) At the brand new middle school lockers open with PIN numbers instead of combination locks. PIN numbers consist of three one digit numbers and digits can repeat. This year all the 7th grade PIN numbers begin with 3, 4 or 5, but the other two can be any number from 0 up to 9. How big does the 7th grade class have to be to guarantee at least two students have the same PIN numbers?
  1. There are 3 ways to pick the first PIN digit
  2. The other 2 digits are 0-9 which is 10 digits each
  3. The maximum number of PIN numbers is 3x10x10 = 300, so the 7th grade class must be 301 to guarantee repetition of the PIN numbers.