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Grade 9 · Math · Algebra 1

9th grade The counting principle, and are these events independent? worksheet

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The counting principle, and are these events independent?

Grade 9 Math Worksheet

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  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?

  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?

  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?

  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?

  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?

  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. 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. 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. 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. 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
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9th grade The counting principle, and are these events independent? worksheet – printable
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About this worksheet

The counting principle, and are these events independent?. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can count the outfits, menus or codes by multiplying the choices at each step, even with no repeats, an "or", or a rule.
  • I can test whether two events are independent by checking whether P(A) × P(B) = P(A and B).
  • I can work back to a missing count or a missing chance by dividing.

Standard: S-CP.B.9 (+, foundation), S-CP.A.2

Words to know

outcome list: Every possible result, one row or one branch each.

counting principle: One choice from each step: multiply the numbers of choices.

no repeats: A choice used once is gone, so the next step has one fewer.

P(A): The chance that event A happens.

P(A and B): The chance that A and B both happen.

independent: P(A) × P(B) equals P(A and B); one event does not change the other's chance.

Common mistakes

Your child added the numbers of choices instead of multiplying them. Try 2 shirts and 3 caps: list the outfits and count all 6.

Your child stopped before the last step of the count. Ask them to name every step before multiplying.

Your child multiplied choices that are alternatives.

What to say

Ask: does one outcome use both choices, or only one?

Questions parents ask

What is the most common mistake with the counting principle, and are these events independent??

Your child added the numbers of choices instead of multiplying them. Try 2 shirts and 3 caps: list the outfits and count all 6.

Are these 9th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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