Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 8 Probability and Statistics
Problem
Solution
1) In the Carter family most arguments are resolved using the Rock, Paper, Scissor game only they've turned it into a card game and each family member has his/her own deck of 6 cards: 2 rocks, 2 papers, 2 scissors that they have to shuffle before they play. Rock beats scissors, scissors beats paper, and paper beats rock. If two family members are resolving an argument what is the probability they draw identical cards on the first play?
  1. Each person has their own 6-card deck
  2. 2 out of 6 cards in the opponent's deck match the first person's card, so the probability is 1⁄3

2) In Holland all cell phone numbers are 10 digits long. The first two digits are always 06; all the other digits can be any one of 0 through 9. The numbers still available are all those that begin with 061 up through 065 and 068. Rina wants to get a number that ends in 424. What is the probability that she will get her wish if numbers are assigned randomly?
  1. There are 6 prefixes: 061,062, 063, 064,065 and 068
  2. Other than the prefixes and Rina's desired last 3 digits, there are 4 unspecified digits which can be any of the digits from 0-9.
  3. With the last 3 digits, the total number of possible phone numbers is 6x107
  4. So the number of numbers that satisfy Rina is 6x104
  5. Therefore, the probability she will get her phone number suffix is 6x104 / 6x107 = 10-3 = 1⁄1000
3) The average for a test in Mrs. Dout's class is only 55 out of 100. Mrs. Dout offers to increase each test score by half of the points they missed if they completely solve the exit task she has for them that day. All students complete the exit task. What is the new average for the test?
  1. Half of the missed points =
    (100 - 55) ÷ 2 = 22.5
  2. The new average is 55 + 22.5 = 77.5

Problem
Solution
4) Mrs. Smith keeps two reward envelopes in her desk. They have colored red, blue, and orange squares in them and each color corresponds to a prize. Envelope A has 3 red, 4 blue, and 1 orange square. Envelope B has 2 red, 2 blue, and 2 orange squares. The only way to get the prize is to give the correct probability for the color you draw. Mrs. Smith pulls out a reward envelope at random and Eric selects a square from that envelope at random. What probability should he tell her if he draws a red? Express your answer as a fraction in lowest terms.
  1. Envelope A: The probability of choosing this envelope is 1/2 and once you choose it, you have a 3⁄8 probability of getting a red for a combined probability of 3⁄16
  2. Envelope B:The probability of choosing this envelope is 1/2 and once you choose it, you have a 1⁄3 probability of getting a red for a combined probability of 1⁄6
  3. The probability of getting a red from A or B is the sum of their probabilities =
    (3⁄16 + 1⁄6) = 17⁄48

5) Data was collected to compare car weight to fuel efficiency and then displayed on the graph shown. Use the trend line to predict the difference in the weight of two cars whose fuel efficiencies differ by 5 mpg.
 Pick any 2 vehicle weights whose mileages differ by 5 mpg.
For example, 4000 pounds (mpg = 15) and
3400 pounds (mpg = 20).
The difference of these weights is
4000 - 3400 = 600 pounds