The counting principle, and are these events independent?

Answer Key - Grade 9 Math Worksheet

Circle the best answer, or write your answer on the line.

1. 1.

   Jun's lock has 4 places with the same number of symbols s in every place. The lock has 1,296 codes. How many symbols are in each place?

   6
2. 2.

   Zee's lock has 3 dials with 7 symbols each. How many more codes are possible when symbols may repeat than when no symbol may repeat?

   133
3. 3.

   A 4-letter code uses the letters A to H, with no letter used twice, and must start with a vowel (A or E). Milo counts the codes. How many are possible?

   420
4. 4.

   A code is 2 letters from A to E (repeats allowed), then 1 digit from 0 to 9. Tala guesses one code at random. What is the chance it is right?

   1/250
5. 5.

   Ari's 3-digit code uses digits 0 to 9 with no digit repeated, and the first digit is not 0. Ari guesses one code at random. What is the chance it is right?

   1/648
6. 6.

   Zee rolls two fair number cubes. A is “the first shows an even number” and B is “the sum is 7”. Are A and B independent?

   • Not independent: 18/36 and 6/36 are not equal
   • Not independent: the sum depends on the first cube
   • Independent: 18/36 × 6/36 = 3/36, and 3 pairs are in both
   • Not independent: 18/36 + 6/36 is not 3/36
7. 7.

   Milo rolls two fair number cubes. A is “the sum is 8” and B is “the first shows an even number”. Are A and B independent?

   • Not independent: 5/36 × 18/36 = 5/72, but P(A and B) = 3/36
   • Independent: the two cubes do not affect each other
   • Independent: P(A and B) is smaller than both P(A) and P(B)
   • Independent: 3 of the 5 sums of 8 have an even first cube, about half
8. 8.

   Esme shuffles cards numbered 1 to 5 and draws two without looking; the first is not put back. A is “first is even” and B is “second is even”. Are A and B independent?

   • Independent: the two draws are different cards
   • Not independent: 8/20 × 8/20 = 4/25, but P(A and B) = 2/20
   • Independent: each draw is even 2 of 5 times, so P(A and B) = 4/25
   • Independent: P(A) and P(B) are equal
9. 9.

   Noah picks one card at random from cards numbered 1 to 20. A is “a multiple of 4” and B is “a multiple of 5”. Are A and B independent?

   • Not independent: multiples of 4 and 5 share the card 20
   • Not independent: 5/20 + 4/20 is not 1/20
   • Not independent: 1/20 is smaller than both 5/20 and 4/20
   • Independent: 5/20 × 4/20 = 1/20, and only 20 is in both
10. 10.

    Tala picks one tile at random from tiles numbered 1 to 10. A is “a multiple of 2” and B is “a multiple of 3”. Are A and B independent?

    • Independent: 5/10 and 3/10 are close to each other
    • Not independent: 5/10 × 3/10 = 3/20, but P(A and B) = 1/10
    • Independent: 2 and 3 share no factor
    • Independent: 1/10 is smaller than both 5/10 and 3/10