Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Probability and Statistics
Problem
Solution
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 A and E
  2. There are 2 possibilities for the first letter
  3. There are 4 possibilities for the second letter (B,C,D,F)
  4. After that, there are 3,2, and 1 choices for the 3rd, 4th and 5th letters.
  5. The total is 2x4x3x2x1 = 48 possible 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 6-sided dice is 6x6x6 = 216
  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 4 or less and there 6 of those
    • Third die is a 2:
      The sum of the other 2 dice must be 3 or less and there are 3 of those
    • Third die is a 3:
      The sum of the other 2 dice must be 2 and there is only 1 combination (1+1)
  4. So we have 6+3+1 = 10 possibilities out of 216 =
    10⁄216 = 5⁄108

Problem
Solution
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).
Event "A" is the probability of choosing bag 1 and getting a red marble.
Event "B" is the probability of choosing bag 2 and getting a red marble.
  1. Event A: The probability of Josiah choosing bag one and getting a red marble is 1⁄2 x 1⁄5 = 1⁄10
  2. Event B: The probability of Josiah choosing bag 2 and getting a red marble is 1⁄2 x 4⁄6 = 1⁄3
  3. The sum of these 2 probabilities is
    p(A or B) = p(A) + p(B):
    1⁄10 + 1⁄3 = 3⁄30 + 10⁄30 = 13⁄30
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:
    53   57   60   61   63   65   70   71
  1. The mean of these numbers is 62.5
  2. The 2 added numbers must add to 150 to make the mean 65
  3. One of these 2 numbers must be the highest value and the other must equal either the 63 or the 65. Examine the 2 cases:
    • The lower of the 2 numbers added is 65. Then the higher number is 150 - 65 = 85, making the entire list:
          53   57   60   61   63 65   65   70   71   85
      This doesn't work, because the median is
      (63+65)/2 = 64
    • The lower of the 2 numbers added is 63. Then the higher number is 150 - 63 = 87, making the entire list:
          53   57   60   61   63 63   65   70   71   87
      This works!
  4. The 2 added numbers are 63 and 87
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 7 letters in ARRANGE and 2 of them are repeated (the R and the A)
  2. The number of ways to arrange the letters in ARRANGE is
        7! / (2! x 2!) = 7! / 4 = 5040/4 = 1260