Applying Complementary Events In Problem Solving Quiz

1) A student guesses on a 4-question true/false quiz. What is the probability they get at least one question wrong?

(1/2)⁴

1 − (1/2)³

(1/2)³

1 − (1/2)⁴

At least one wrong = 1 − (all correct) = 1 − (1/2)⁴.

Explanation

At least one wrong = 1 − (all correct) = 1 − (1/2)⁴.

2) A survey shows 40% of people prefer tea to coffee. If two people are chosen independently at random, what is the probability that at least one prefers coffee?

(0.6)²

0.4 × 0.6

1 − (0.6)²

1 − (0.4)²

Coffee = 0.6. At least one coffee = 1 − (both tea) = 1 − (0.4)².

Explanation

Coffee = 0.6. At least one coffee = 1 − (both tea) = 1 − (0.4)².

3) In a raffle, 200 tickets are sold and 5 are winners. If you buy one ticket, what is the probability that you do not win?

5/200

195/200

7/200

100/200

Not winning = 1 − 5/200 = 195/200.

Explanation

Not winning = 1 − 5/200 = 195/200.

4) A game wheel has equally likely outcomes labeled A, B, C, D, E. What is the probability of not landing on A in three independent spins?

(4/5)³

1 − (4/5)³

1 − (1/5)³

(1/5)³

Not A per spin = 4/5 → three spins = (4/5)³.

Explanation

Not A per spin = 4/5 → three spins = (4/5)³.

5) A student will attempt a skill until they succeed. Each attempt has a 0.6 chance of success, independent of previous attempts. What is the probability they fail both of their first two attempts?

1 − 0.6²

1 − 0.4²

0.4²

0.6²

Fail per attempt = 0.4 → two fails = 0.4 × 0.4 = 0.4².

Explanation

Fail per attempt = 0.4 → two fails = 0.4 × 0.4 = 0.4².

6) A web server has a 2% chance of being down on any given minute, independently each minute. What is the probability that it is up for all 10 minutes of a check?

1 − 0.02¹⁰

1 − 0.98¹⁰

0.02¹⁰

0.98¹⁰

Up per minute = 0.98 → all 10 minutes = 0.98¹⁰.

Explanation

Up per minute = 0.98 → all 10 minutes = 0.98¹⁰.

7) A standard deck has 52 cards. What is the probability that a single card drawn is not a heart?

1/4

3/4

13/52

1 − 3/4

Hearts = 13/52 → not heart = 39/52 = 3/4.

Explanation

Hearts = 13/52 → not heart = 39/52 = 3/4.

8) A box has 12 bulbs, 3 defective and 9 good. One bulb is selected at random. What is the probability that it is not defective?

9/12

5/12

3/12

1/12

Not defective = 9/12.

Explanation

Not defective = 9/12.

9) The probability that it rains on any given day is 0.3, independently day to day. What is the probability that it doesn’t rain at all over a 3-day weekend?

1 − 0.3³

0.7³

1 − 0.7³

0.3³

No rain for 3 days = (0.7)³.

Explanation

No rain for 3 days = (0.7)³.

10) A die is rolled twice. What is the probability of not rolling a 6 on either roll?

1 − (1/6)²

(5/6)²

(1/6)²

1 − (5/6)²

Not 6 per roll = 5/6 → both rolls = (5/6)².

Explanation

Not 6 per roll = 5/6 → both rolls = (5/6)².

11) What is the probability of not landing on Red in one spin?

3/8

5/8

1/2

1/8

Not Red = 1 − 3/8 = 5/8.

Explanation

Not Red = 1 − 3/8 = 5/8.

12) A password requires choosing a digit from 0–9. What is the probability that a randomly chosen digit is not a prime number? (Prime digits: 2, 3, 5, 7)

1/10

3/10

6/10

8/10

Non-prime digits = 10 − 4 = 6 → 6/10.

Explanation

Non-prime digits = 10 − 4 = 6 → 6/10.

13) A bag contains 5 red, 4 blue, and 1 yellow marble. One marble is drawn at random and replaced. What is the probability that in three draws you never pick a yellow marble?

(9/10)³

1 − (9/10)³

1 − (1/10)³

(1/10)³

Not yellow each draw = 9/10 → three draws = (9/10)³.

Explanation

Not yellow each draw = 9/10 → three draws = (9/10)³.

14) In a class of 30 students, 18 play soccer. If a student is chosen at random, what is the probability they do not play soccer?

18/30

1 − 12/30

12/30

0.6

Not soccer = (30 − 18)/30 = 12/30.

Explanation

Not soccer = (30 − 18)/30 = 12/30.

15) A fair coin is flipped 5 times. What is the probability of getting at least one tail?

(1/2)⁵

1 − (1/2)⁵

1 − (1/2)⁴

(1/2)⁴

At least one tail = 1 − (all heads) = 1 − (1/2)⁵.

Explanation

At least one tail = 1 − (all heads) = 1 − (1/2)⁵.

16) What is the probability that in two spins, you do not land on Yellow or Green in either spin?

1 − (3/8)²

1 − (5/8)²

(5/8)²

(3/8)²

Not (Y or G) per spin = 1 − (1/8+2/8) = 5/8 → (5/8)².

Explanation

Not (Y or G) per spin = 1 − (1/8+2/8) = 5/8 → (5/8)².

17) In four spins, what is the probability of landing on Red at least once?

(3/8)⁴

(5/8)⁴

1 − (3/8)⁴

1 − (5/8)⁴

At least one Red = 1 − (no Red)⁴ = 1 − (5/8)⁴.

Explanation

At least one Red = 1 − (no Red)⁴ = 1 − (5/8)⁴.

18) What is the probability of landing on Green or Blue at least once in two spins?

1 − (1/2)²

(1/2)²

1 − (1/2)³

(3/8)²

P(G or B) = 2/8 + 2/8 = 1/2 → at least once = 1 − (1 − 1/2)².

Explanation

P(G or B) = 2/8 + 2/8 = 1/2 → at least once = 1 − (1 − 1/2)².

19) What is the probability that in three spins, the spinner never lands on Yellow?

(1/8)³

1 − (1/8)³

(7/8)³

1 − (7/8)³

Not Yellow each time = (7/8)³.

Explanation

Not Yellow each time = (7/8)³.

20) What is the probability of landing on Blue at least once in two spins?

1 − (6/8)

(1/4)²

1 − (1/4)²

1 − (3/4)²

At least one Blue = 1 − (no Blue)² = 1 − (3/4)².

Explanation

At least one Blue = 1 − (no Blue)² = 1 − (3/4)².

Applying Complementary Events in Problem Solving Quiz

1) A student guesses on a 4-question true/false quiz. What is the probability they get at least one question wrong?

2)

What first name or nickname would you like us to use?

2) A survey shows 40% of people prefer tea to coffee. If two people are chosen independently at random, what is the probability that at least one prefers coffee?

3) In a raffle, 200 tickets are sold and 5 are winners. If you buy one ticket, what is the probability that you do not win?

4) A game wheel has equally likely outcomes labeled A, B, C, D, E. What is the probability of not landing on A in three independent spins?

5) A student will attempt a skill until they succeed. Each attempt has a 0.6 chance of success, independent of previous attempts. What is the probability they fail both of their first two attempts?

6) A web server has a 2% chance of being down on any given minute, independently each minute. What is the probability that it is up for all 10 minutes of a check?

7) A standard deck has 52 cards. What is the probability that a single card drawn is not a heart?

8) A box has 12 bulbs, 3 defective and 9 good. One bulb is selected at random. What is the probability that it is not defective?

9) The probability that it rains on any given day is 0.3, independently day to day. What is the probability that it doesn’t rain at all over a 3-day weekend?

10) A die is rolled twice. What is the probability of not rolling a 6 on either roll?

11) What is the probability of not landing on Red in one spin?

12) A password requires choosing a digit from 0–9. What is the probability that a randomly chosen digit is not a prime number? (Prime digits: 2, 3, 5, 7)

13) A bag contains 5 red, 4 blue, and 1 yellow marble. One marble is drawn at random and replaced. What is the probability that in three draws you never pick a yellow marble?

14) In a class of 30 students, 18 play soccer. If a student is chosen at random, what is the probability they do not play soccer?

15) A fair coin is flipped 5 times. What is the probability of getting at least one tail?

16) What is the probability that in two spins, you do not land on Yellow or Green in either spin?

17) In four spins, what is the probability of landing on Red at least once?

18) What is the probability of landing on Green or Blue at least once in two spins?

19) What is the probability that in three spins, the spinner never lands on Yellow?

20) What is the probability of landing on Blue at least once in two spins?