Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 7 Probability and Statistics
Problem
Solution
1) A basketball team was practicing their shots. Each player took five shots. Michael made 100% of his shots, while Steve only made 40%. If John made 3 shots, David made 2 and Brett made 1, what is the team's average of all the shots taken? (5 + 2 + 3 + 2 + 1) / 25 = 13/25 = 52%

2) Most Washington license plates consist of a set of three numbers followed by a set of three letters. A witness sees a suspicious car and reports it to the police. He says the first two numbers were 4 and 6 but he missed the last one and the first letter was C. How many possible license plates does that describe? 		There is 1 number left and 2 letters left, so the number of possible license plates is 10x26x26 = 6760

3) A dart hits the dartboard shown at random. The board is a square and the horizontal lines are equally spaced. Find the probability of the dart landing in the shaded region. 		You need to find the total area of the two shaded areas. Use 1 for the square side length
  1. The large lower triangle and the shaded triangle that is inside of it are similar because they have parallel bases.
  2. The height of the large triangle is .5 and the height of the included triangle is 1⁄2 x 1⁄3 = 1⁄6
  3. Therefore the ratio of their side lengths is 1⁄2 ÷ 1⁄6 = 3
  4. The base of the small triangle is, therefore, 1⁄3
  5. The area of the 2 small shaded triangles and the probability of the dart hitting them) is
        1⁄3 x 1⁄6 = 1⁄18

Problem
Solution
4) A company manufactures washers that are supposed to be 2 mm thick. In reality they will accept washers that are within 1 mm of the 2 mm target. The company buys a new machine to produce these washers. Out of the first 100 washers it produces, the median thickness is 2.5 mm, the range is 3.9 mm, the first quartile median is 1.8 mm, and the third quartile median is 3 mm. The thickest washer had a thickness of 5 mm. From this data at least how many washers of the 100 washers did not meet the company's requirements?
  1. If the range is 3.9 mm and the thickest washer is 5 mm, the the minimum thickness was 5 - 3.9 = 1.1 mm.
  2. The median is 2.5 mm, so all the washers below the median are acceptable.
  3. If the median of the third quartile (the one just above the whole set median) was 3 mm (the upper limit) it is possible that all the third quartile washers were acceptable if all of them above the median were 3 mm thick (unlikely)
  4. This means that all the washers in the 4th quartile were unacceptable (their thicknesses were above 3 mm.), therefore the least number of washers that did not meet the company's criteria was 25 washers of the 4th quartile.
5) Joan and David invent a probability game for a homework project that has players flip a fair coin and roll a fair 6 sided number cube with the numbers 1 through 6 on it. David moves ahead on the game board as many spaces as the number that comes up on the number cube when the coin lands on heads. Joan moves ahead three spaces when the coin lands on tails no matter what number comes up on the cube. After 2 rolls and flips what's the probability that David will have moved 8 spaces?
  1. The number of possible outcomes is
    (2x6) x (2x6) = 144
  2. The only outcomes that can produce a move of 8 spaces are when David rolls a head both times, which is 1⁄4 of the possibilities = 36 possibilities
  3. Of those 36 possibilities here are the dice combinations that produce 8 spaces:
    1. 2+6
    2. 3+5
    3. 4+4
    4. 5+3
    5. 6+2
  4. So the probability of David moving 8 spaces is 5/144