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.

   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
4. 4.

   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
5. 5.

   Luz's lock has 3 dials, each showing 0 to 9. A new lock adds 2 more dials. By what fraction is the chance of a random correct guess multiplied?

   1/100
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: the sum depends on the first cube
   • Not independent: 18/36 + 6/36 is not 3/36
   • Independent: 18/36 × 6/36 = 3/36, and 3 pairs are in both
   • Not independent: 18/36 and 6/36 are not equal
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?

   • Independent: 3 of the 5 sums of 8 have an even first cube, about half
   • 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)
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.

   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: 6 and 12 are both multiples of 3 and even
   • Not independent: 4/12 + 6/12 is not 2/12
   • Not independent: 4/12 and 6/12 are not equal
   • Independent: 4/12 × 6/12 = 2/12, and 6 and 12 are in both
10. 10.

    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?

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