Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 6 Probability & Statistics
Problem
Solution
1) You are the manager of the girls' basketball team. The championship game is coming up, and the coach has asked you to determine from his statistics which of his top three players has the best record for free throws. Here is the statistical information that you have:
Joy made 21 out of her last 25 free throws.
Mary made 17 out of her last 20 free throws.
Nancy made 12 out of her last 15 free throws.
Tell the manager the order of his top 3 players: who is first, who is second, and who is third? 		1. Compute the percentage of successful free throws for each girl.
2. Order them from best down.
    Joy 21/25 = 84%
    Mary 17/20 = 85%
    Nancy 12/15 = 80%
3. Mary is first, Joy is second and Nancy is third.

2) A set of 5 whole numbers all more than 0 but less than 10 has a mean of 5, a mode of 2, and a median less than the mean. What is the set of numbers?		 1. OK, if 2 is the mode then there are at least 2 of them and no more than 1 of any other number.
2. For the mean to be 5, the 5 numbers must add up to 25, so if there are two 2s then the other 3 numbers must add to 21
3. If the median is less than the mean,
    then it must be either a 2, 3 or a 4.
4. Examine the possibilities for the last 2 numbers given the two 2s and these 3 possibilities for the median.
    2 2 2 ___ ___. The last 2 numbers add to 19. Impossible.
    2 2 3 ___ ___. The last 2 numbers add to 18. Impossible.
    2 2 4 ___ ___. The last 2 numbers add to 17.
    8 and 9 work = 17!
5. The set of numbers is { 2 2 4 8 9 }

Problem
Solution
3) Nora is considering inventing a language that consists of only three letter words. The words will all be made up of one vowel in the middle and 2 consonants so that they can be pronounced. She has decided to treat y as a vowel. How many different three letter "words" could be in her language? So Nora's language is of the form "C V C" where
  C is a consonant and V is a vowel.
1. Count up the number of vowels, including y =
    a,e,i,o, u and y = 6 vowels
2. Count up the number of consonants as 26 minus this number =
    26 - 6 = 20
3. Then Nora's language has 20 x 6 x 20 = 2400 words.
4) Greg tosses 4 pennies in the air and they come up 3 heads and 1 tail. Express the probability of this happening as a reduced fraction. There are a total of 16 possibilities: 2 for each coin.
There are 2 methods to solve this problem:
Method 1: brute force:
The ones that are 3 heads and a tail have an asterisk by them:
1. H H H H
2. H H H T *
3. H H T H *
4. H H T T
5. H T H H *
6. H T H T
7. H T T H
8. H T T T
9. T H H H *
10. T H H T
11. T H T H
12. T H T T
13. T T H H
14. T T H T
15. T T T H
16. T T T T
There are 4 out of 16 which is a probability of 1/4 		Method 2: Analysis:
For each possibility there are 4 coin possitions. The ones that have 3 heads and 1 tail are:
    H H H T
    H H T H
    H T H H
    T H H H
This makes 4 out of the 16 possible combinations for a probability of 1/4
5) A basketball team won 30 of the first 52 games it played. If this is a statistically accurate reflection of the team's abilities for the entire season, how many games are they likely to win of the next 18 games they play? 1. Compute the teams winning percentage =
    30/52 = 57.7%
2. Multiply this by the number of games they will now play =
    They will win 18 x 0.577 = 10.3 = 10 games.