Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Probability and Statistics
Problem
Hint
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. Convert both chances to fractions and multiply:


= _____________

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? 		Compute the number of students that can't do without toilet paper and zippers:
  1. Toilet paper: _____ students

  2. Zippers: _____ students
Any overlap in the sum of these number of students and the total number of students is the minimum amount of overlap between these 2 categories
= _____ students.

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?
  1. Using the percentages, compute the number of students in each ethnic group:
    White: _______
    Hispanic: _______
    Asian: _______
    African American: _______
    Native American: _______
    Other: _______

  2. The share of the population shared by the African American, Native American and other ethnicities is a total of ______ students
  3. Turn the population percentage changes into a multiplicative factor and apply the changes to the top 3 categories:
    White: ______ students

    Hispanic: ______

    Asian: ______
  4. The total for these 3 ethnicities: ________ current students.
  5. Since the total student population remains the same, the other 3 ethnicities share ______ students.

  6. This means those 3 ethnicities went up collectively by ____%
  7. The Native American share of this new population is ________ 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? OK, you're going to buy an additional number of raffle tickes (call that X) to get your total probability to 1/5. You currently have 10 tickets out of a total of 150 and you must add this X to both the numerator (the number you have) and the denominator (the total number of tickets) to have that fraction equal 1/5. You should be able to write this equation and solve for X:




X = _____ additional 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 ______

  3. The probability that that birthday is not C's birthday is ______

  4. So the combined product of these 2 terms is the probability that A & B have the same birthday but not C:


  5. There are 3 of these combinations, so the combined probability is 3 times this probability


    = _________