Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Probability and Statistics
Problem
Hint
1) Boitumelo can't remember his passcode. He remembers that it has five letters and that it begins and ends with a vowel. He also remembers that no letter is repeated in his code. The picture below shows the keypad that he enters his code on (only the letters A through F). How many possibilities are there for his passcode?
  1. The vowels are ___________
  2. There are ___ possibilities for the first letter
  3. There are ____ possibilities for the second letter (B,C,D,F)
  4. After that, there are ___, ___, and ____ choices for the 3rd, 4th and 5th letters.


  5. The total is ______ combinations.
2) Three six-sided dice are rolled. What is the probability that the sum of the 3 dice will be less than 6? Express you answer as a fraction in lowest terms. The table to the right shows the sums of 2 dice.
  1. The total number of combinations of 3 dice are ______
  2. The sum must be 5 or less
  3. There are 3 possibilities for the 3rd die:
    • Third die is a 1:
      The sum of the other 2 dice must be ____ or less and there ____ of those
    • Third die is a 2:
      The sum of the other 2 dice must be ____ or less and there are ____ of those
    • Third die is a 3:
      The sum of the other 2 dice must be ___ and there is _____ of those.
  4. So we have ____ possibilities out of ____ =


    _________________

Problem
Hint
3) There are two bags. The first bag has 1 red marble, 2 blue marbles and 2 yellow marbles. The second bag has 4 red marbles, 1 blue marble, and 1 yellow marble. Josiah picks a bag at random. He then draws a marble at random from that bag. What is the probability that he draws a red marble?
Express your answer as a fraction in lowest terms. 		We have the probability of one of 2 events (A or B):
The probability of A or B is:
    p(A or B) = p(A) + p(B).
  1. The probability of Josiah choosing bag one and getting a red marble is ____________________

  2. The probability of Josiah choosing bag 2 and getting a red marble is ________________

  3. The sum of these 2 probabilities is


    __________________
4) What two numbers would you add to this list to make the median the same as the mode and to make the mean 65?
    60, 53, 71, 65, 61, 63, 57, 70
 First, rewrite the list, putting the numbers in order:


  1. The mean of these 8 numbers is _____
  2. The 2 added numbers must add to _____ to make the mean 65
  3. One of these 2 numbers must match one of the numbers in the middle of the list to make the median the same as the mode. The other makes the total of the 2 added numbers sum to 150.
  4. Try different combinations to satisfy the requirements.
  5. The 2 added numbers are ____ and ____.
5) In how many unique ways can you arrange the letters of the word "ARRANGE"? This is the permutation of the letters in a word with repeated letters. This involves factorials
  1. There are ___ letters in ARRANGE and ___ of them are repeated.
  2. The number of ways to arrange the letters in ARRANGE is


        ____________________