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.

   Esme's mailbox code has 6 places with the same number of symbols s in every place. The mailbox has 729 codes. How many symbols are in each place?

   3
2. 2.

   A tree's first level has 4 ends, and every end splits into 4 new ends on each next level. Pia wants to know how many ends level 5 has.

   1024
3. 3.

   Kira's 3-digit code uses the digits 1 to 7, with no digit used twice, and the code must be even. How many codes are possible?

   90
4. 4.

   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
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.

   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: 3 of the 5 sums of 8 have an even first cube, about half
   • Independent: P(A and B) is smaller than both P(A) and P(B)
   • Independent: the two cubes do not affect each other
7. 7.

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

   Ren's spinner has 12 equal parts numbered 1 to 12. A is “a multiple of 3” and B is “an even number”. Are A and B independent?

   • Not independent: 4/12 and 6/12 are not equal
   • Not independent: 4/12 + 6/12 is not 2/12
   • Independent: 4/12 × 6/12 = 2/12, and 6 and 12 are in both
   • Not independent: 6 and 12 are both multiples of 3 and even
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: 1/20 is smaller than both 5/20 and 4/20
   • Not independent: 5/20 + 4/20 is not 1/20
   • 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?

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