Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Probability and Statistics
Problem
Solution
1) Roy is looking forward to making some good money once he's in high school. He does an informal survey of his older brother's friends. He puts the results of his survey in a table shown below. Compute the mean and median hourly wage for Roy's informal survey. Is Roy more likely to get a job that pays close to the mean hourly wage or to the median hourly wage? Explain.
  1. Compute the median:
    There are 20 measurements, an even number, so the median is the average of the 10th and 11 measurements. They are both $9
  2. The mean is the sum of all the wages divided by 20 =
    194/20 = $9.70
  3. Roy is most likely to get the median because the data includes a big outlier, the $20 wage which affected the mean. Without that outlier, the mean would be much closer to the median.

2) Here's a game with one standard six sided die with the numbers 1 - 6. In at most three rolls of the die be the one to get as close to 10 as possible without going over. If you roll a 4 on the first roll, what's the probability you will go over 10 in the next two?
  1. If you roll a 4 on the first die, then any roll over 6 will go over 10.
  2. If you look at the sum of 2 dice table to the right, there are 21 2-dice combinations that result in 7 or more, so the probability that you will go over is 21/36 = 7/12

Problem
Solution
3) College bound high school students often take a foreign language. One school has a senior class of 255 students. 178 are college bound and they all take either French or Spanish. 97 students have taken at least 3 years of high school French, and 123 have taken at least 3 years of high school Spanish. No student has taken at least 3 years of both languages. If a student is selected at random, what's the probability they've had at least three years of French or Spanish but are not college bound?
  1. There are 255 - 178 = 77 students that are not college bound
  2. There are a total of 123 + 97 = 220 students that have taken at least 3 years of French or Spanish and 178 of them are college bound, leaving 220 - 178 = 42 that have taken 3 years of a foreign language and are not college bound.
  3. The probability that a non-college-bound student takes 3 years of a foreign language is therefore
    42⁄77
4) A coin is weighted so that it comes up heads twice as often as tails. If it comes up heads 5 times in a row, what's the probability it will come up tails on the next toss? What's happened in the past doesn't affect what happens in the future for independent probabilities, so it is 1 tail and 2 heads for a probability of getting a tail of 1⁄3

5) Gavin had a GPA of 2.9 going into his last semester of 8th grade. He would like to bring it up to a 3.0 or higher. Each semester he has taken 6 classes and each one is worth 1 credit. His middle school is 6th through 8th for a total of 6 semesters. What's the lowest his last semester's GPA can be to get a cumulative GPA of 3.0?
  1. Gavin has averaged a GPA of 2.9 for 5 semesters.
  2. Using G for his last semester GPA, the equation for the mean is:
    (2.9x5 + G) = 3.0
        6
    2.9x5 + G = 18
    G = 18 - 2.9x5 = 18 - 14.5 = 3.5