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.

   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
2. 2.

   A club with n members picks a president and a different treasurer, so it has n(n − 1) ways. How many more ways are there when Ren's club grows from 12 to 15 members?

   78
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.

   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
7. 7.

   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
8. 8.

   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: P(A and B) is smaller than both P(A) and P(B)
   • Independent: the two cubes do not affect each other
   • 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
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
   • Independent: 4/12 × 6/12 = 2/12, and 6 and 12 are in both
   • Not independent: 4/12 + 6/12 is not 2/12
   • Not independent: 4/12 and 6/12 are not equal
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