The addition rule, and the general multiplication rule

Grade 11 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   P(A) = 0.55 and P(B) = 0.65. Ari wonders whether A and B could be mutually exclusive. Which statement is right?

   • They cannot be: P(A) + P(B) = 1.2, so they overlap by at least 0.2
   • They overlap by exactly 0.2, so P(A and B) = 0.2
   • They overlap by at most 0.2, since the total is 1.2
   • They can be: each chance is under 1, so they may avoid each other
2. 2.

   P(A) = 0.3 and P(B) = 0.4, and A and B cannot both happen. Bo claims they are therefore independent. Is that right?

   • Yes: events that cannot happen together do not affect each other
   • Yes: P(A or B) = 0.3 + 0.4 = 0.7, which is the independent rule
   • No: given A, B cannot happen, so P(B | A) = 0, not P(B) = 0.4
   • No: P(B | A) = 0.3 × 0.4 = 0.12, which differs from 0.4
3. 3.

   Jun knows P(A | B) > P(A). Which statement about P(B | A) and P(B) must be true?

   • P(B | A) > P(B) just when P(A) = P(B)
   • P(B | A) = P(B), since just A depends on B
   • P(B | A) > P(B) just if the events are independent
   • P(B | A) > P(B) too, since P(A and B) is above P(A) × P(B)
4. 4.

   A bag has 3 red and 5 blue marbles. Kira draws three without replacement. What is P(exactly two red)?

   • 3/8 × 2/7 = 3/28, the two reds alone
   • 3 × (3/8 × 2/7 × 5/6) = 15/56
   • 3 × (3/8)² × 5/8 = 135/512
   • 3/8 × 2/7 × 5/6 = 5/56, red, red, blue alone
5. 5.

   A bag has 3 red and 5 blue marbles. Luz draws two without replacement and sees that the second is blue. What is P(the first was red)?

   • (3/8 × 5/7) ÷ (5/8) = 3/7
   • 5/7, the chance of blue after red
   • 15/56, the chance of red then blue
   • 3/8, the chance the first is red
6. 6.

   A lot of 20 parts has 3 defective. Noah picks two without replacement. What is P(at least one defective)?

   • 1 − 17/20 × 16/19 = 27/95
   • 3/20 × 2/19 = 3/190, both defective
   • 1 − (17/20)² = 111/400
   • 3/20 + 3/19 = 117/380
7. 7.

   Pia claims P(A and B) = P(A) × P(B) for any events. A tile is drawn from 1 to 12. Which pair shows the claim fails?

   • A is a multiple of 3, B is odd, since both contain 3 and 9
   • A is odd, B is less than 7, since both contain 1, 3 and 5
   • A is even, B is a multiple of 3, since both contain 6 and 12
   • A is even, B is less than 4, since P(both) = 1/12 but 1/2 × 1/4 = 1/8
8. 8.

   A lot of 12 parts has 4 defective. Ren tests three without replacement. What is P(none defective)?

   • 4/12 × 3/11 × 2/10 = 1/55, three defective parts
   • (8/12)³ = 8/27
   • 8/12 × 7/11 × 6/10 = 14/55
   • 8/12 × 7/11 × 6/11 = 28/121
9. 9.

   P(A) = 0.5, P(B) = 0.4 and P(A or B) = 0.9. Tala decides what must be true about A and B. Which statement fits?

   • Independent, since P(A) + P(B) = P(A or B)
   • They overlap by 0.1, since 1 − 0.9 = 0.1
   • Mutually exclusive, so P(B | A) = 0 and they are not independent
   • Nothing can be said without P(A and B)
10. 10.

    P(A) = 0.3. If A happens, B has chance 0.5; if A does not, B has chance 0.2. Zee finds P(A or B). Which is right?

    • 0.3 + 0.7 × 0.2 = 0.44
    • 0.3 + 0.29 − 0.3 × 0.29 = 0.503
    • 0.3 + 0.29 = 0.59, adding P(B) = 0.29
    • 0.3 + 0.2 = 0.5