Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Probability and Statistics
Problem
Solution
1) 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 nearest hundredth. 3⁄4 x 3⁄4 = 9⁄16

2) 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.

Problem
Solution
3) A large university enrolls a student population of 85,000 students each year. The ethnic make up of the university population for last year is shown on the pie chart. This year the Asian student population percentage increased by 4% and the Hispanic population percentage by 2%. The white population percentage decreased by 9%. The other three ethnic groups each changed by the same percentage. How many Native American students are there this year? Existing population numbers:
  1. Starting population numbers:
    White: .72 x 85,000 = 61,200
    Hispanic: .11 x 85,000 = 9350
    Asian: .08 x 85,000 = 6800
    African American: .05 x 85,000 = 4250
    Native American: .03 x 85,000 = 2550
    Other: .01 x 85000 = 850
  2. The share of the population shared by the African American, Native American and other ethnicities is 4250 + 2550 + 850 = 7650 students
  3. Apply population changes:
    White: 61,200 x .91 = 55692
    Hispanic: 9350 x 1.02 = 9537
    Asian: 6800 x 1.04 = 7072
  4. The total for these 3 ethnicities: 55692 + 9537 + 7072 = 72,301
  5. The other 3 ethnicities share 85000 - 72301 = 12,699 students
  6. This means those 3 ethnicities went up collectively by 12699/7650 = 1.66 = 166%
  7. The Native American share of this advance was 1⁄3 =
  8. The new Native American population is 2550 x 1.66 = 4233 students.
4) The eighth grade at Sonora MS is raising money for the local food bank. Part of their fundraising is selling raffle tickets for a set of 4 tickets to a U2 concert that someone donated. Ryan really wants those tickets. He knows they go for over $50 a ticket. He's bought 10 raffle tickets so far. They're about to stop selling tickets and he finds out that about 150 raffle tickets have been sold. He wants to increase the chance that he will win to 20%. How many more tickets should he buy if no one else buys more tickets?
  1. The equation is:
    10 + x = 1/5
    150 + x
  2. 50 + 5x = 150 + x
  3. 4x = 100
  4. X = 25 tickets

Problem
Solution

5) Three people meet at a party and discover they have birthdays in May. What's the chance that exactly two of their birthdays are all on the same day? May has 31 days. You may leave your answer as an expression or give your answer to the nearest hundredth. 		Call the 3 people A, B and C.
  1. We have the probability of 3 separate things:
    1. A & B have the same birthday but not C
    2. A & C have the same birthday but not B
    3. B & C have the same birthday but not A
  2. The probability that A & B have the same birthday is 1⁄31
  3. The probability that that birthday is not C's birthday is 30⁄31
  4. So the combined product of these 2 terms is the probability that A & B have the same birthday but not C:
    (1⁄31)(30⁄31)
  5. There are 3 of these combinations, so the combined probability is:
    3(1⁄31)(30⁄31) = 90⁄961