Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 7 Probability and Statistics
Problem
Solution
1) At the local middle school math night, one game has a grand prize that everyone wants. To play and win you need to get three 6's. There are two options:
Option 1: roll a regular number cube with the numbers 1 to 6 on it three times and hopefully roll a 6 each time.
Option 2: pull three balls from a bin that contains 18 balls; each ball has one number on it and there are three balls for each number from 1 to 6.

Which option is better and why? Include the probabilities for both options in your answer.
  1. Probability #1: 3 6s with 3 rolls of a single die:
        1⁄6 x 1⁄6 x 1⁄6 = 1⁄216
  2. Probability #2: 3 balls from a bin:
    This is the same as the problem of selecting balls out of a jar without replacing the ball selected, so each pull has one less ball to pull from, so ...
        3⁄18 x 2⁄17 x 1⁄16 =
        1⁄3 x 1⁄17 x 1⁄16 = 1⁄816
  3. Option #1 is the best

Problem
Solution
2) Lena is signing up for classes at her summer camp. She wants to take gymnastics, horseback riding, drawing, and theatre just like her friend Rhoda. Gymnastics and drawing are offered all four sessions. Horseback riding is offered session 2 and 3. Theatre is offered session 1 and 4. Campers take one class a session. She wants the same schedule Rhoda has; Gymnastics, drawing, horseback riding and theatre, in that order. If she is assigned randomly to a schedule that includes her four choices, what is the probability she will get the same schedule as Rhoda? Use G = gymnastics, D = drawing,
T = theatre and H = horseback riding.
The best way to do this is to list the possibilities. Here is a diagram of the 4 sessions, remembering that H is available only in sessions 2 and 3 and Theatre is available only in sessions 1 and 4:
Session 1Session 2Session 3Session 4
T or
G or D		H or
G or D		H or
G or D		T or
G or D

Here are the possibilities (Rhoda's schedule in bold):
Session 1Session 2Session 3Session 4
TH DG
THGD
TGHD
TDHG
GH DT
GD HT
DHGT
DGHT
There are 8 ways to schedule the 4 classes and Rhoda's schedule is one of them, so the probability of Lena getting the same schedule as Rhoda is 1⁄8

Problem
Solution
3) Three cousins all have birthdays that fall on the 27th day of different months and none were born in a leap year. What is the probability of that happening? Express your answer as a fraction, you need not reduce it.
  1. Assume that the first cousin was born on the 27th. Then there are 12/365 27ths in the year that he could have been born on.
  2. If the second cousin was born in a different month, then there are 11 days out of 365 for his/her birthday
  3. That leaves 10 days out of 365 for the last cousin's birthday, so, the probability of this happening is ...
  4. 12⁄365 x 11⁄365 x 10⁄365 = (12 x 11 x 10)⁄3653
4) The Bin Candy Store has candy bins arranged along two walls. Along the first wall the price is $3.95 per pound. Along the second wall the price is $6.50 per pound. On Saturday the store sells 450 pounds from the first wall bins and 200 pounds from the second wall bins. What was the average cost in dollar per pound of candy sold on Saturday? Express your answer to the nearest cent. They sold a total of 650 pounds of candy at a price of ($3.95x450 + $6.50x200) = $3,077.50 for 650 pounds of candy = $4.73 per pound
5) Scientists are monitoring the population of a parrot species in a jungle. One day they catch and tag 50 parrots and release them. A week later they return and catch 50 parrots again and find that 15 of those parrots are tagged. Based only on this data, what is a good initial estimate for the population of this species of parrot in this jungle? Let X be the total population of parrots.
  1. 15/50 of the parrots in the second sample were tagged
  2. This fraction was 50 parrots, so (15/50) X = 50
  3. X = 166.67 = 167 parrots