Year 11 Maths: Probability of Independent and Dependent Events

Understanding Probability

Probability is a way of measuring how likely something is to happen. We can use it to predict the outcomes of events. In this lesson, we will focus on two types of events: independent events and dependent events.

Independent Events

What are Independent Events?

Independent events are events that do not affect each other’s outcomes. This means that the result of one event will not change the result of another event.

Example of Independent Events

Consider the following situation:

• Flipping a Coin: When you flip a coin, it can either land on heads or tails. If you flip the coin again, the first flip doesn’t change the outcome of the second flip. Each flip is independent.

Key Rule for Independent Events

To find the probability of two independent events happening together, you multiply their probabilities:

$$P(A \text{ and } B) = P(A) \times P(B)$$

Example Calculation

If the probability of flipping heads (event A) is $$P(A) = \frac{1}{2}$$ and the probability of rolling a 3 on a die (event B) is $$P(B) = \frac{1}{6}$$ , then the probability of both happening is:

$$P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

Dependent Events

What are Dependent Events?

Dependent events are events where the outcome of one event affects the outcome of another event. This means that knowing the result of one event gives you information about the other event.

Example of Dependent Events

Consider this situation:

• Drawing Cards: If you draw a card from a deck, the outcome changes the situation for the next draw. For example, if you draw an Ace, there are now fewer cards in the deck, and the number of Aces has decreased.

Key Rule for Dependent Events

To find the probability of two dependent events, you find the probability of the first event and then multiply it by the probability of the second event given that the first event has happened:

$$P(A \text{ and } B) = P(A) \times P(B \text{ given } A)$$

Example Calculation

If you draw a card from a standard deck (52 cards) and the probability of drawing an Ace (event A) is:

$$P(A) = \frac{4}{52} = \frac{1}{13}$$

If you then want to find the probability of drawing a second Ace (event B), given that the first card drawn was an Ace, there are now only 51 cards left with 3 Aces remaining:

$$P(B \text{ given } A) = \frac{3}{51}$$

So, the probability of both events happening is:

$$P(A \text{ and } B) = P(A) \times P(B \text{ given } A) = \frac{1}{13} \times \frac{3}{51} = \frac{3}{663} = \frac{1}{221}$$

Tips and Tricks

1. Identify the Events: Always determine if the events are independent or dependent.
2. Use the Right Formula: Make sure to use the multiplication rule for independent events and the conditional probability for dependent events.
3. Practice: The more you practice, the more comfortable you’ll become with these concepts.

Questions

Easy Level Questions (1-20)

1. What is the probability of flipping a coin and getting heads?
2. If you roll a die, what is the probability of getting a number greater than 3?
3. If you flip a coin twice, what is the probability of getting tails both times?
4. You have a bag with 2 red balls and 3 green balls. What is the probability of picking a red ball?
5. If you roll two dice, what is the probability of getting a total of 7?
6. What is the probability of drawing a heart from a standard deck of cards?
7. You flip a coin and roll a die. What is the probability of getting heads and a 5?
8. If you draw a card from a deck, what is the probability of it being a King?
9. What’s the probability of rolling a 4 on a die?
10. If you flip a coin three times, what is the probability of getting at least one head?
11. You have a box with 10 marbles (5 blue, 5 red). What is the probability of picking a blue marble?
12. If you roll a die, what is the probability of not rolling a 6?
13. What is the probability of drawing an Ace from a deck of cards?
14. You flip a coin and get tails. What is the probability of getting tails again on the next flip?
15. If you have 3 green balls and 2 yellow balls, what is the probability of picking a yellow ball?
16. What’s the probability of getting a number less than 4 when rolling a die?
17. What is the probability of drawing a spade from a standard deck of cards?
18. If you flip a coin twice, what is the probability of getting at least one tails?
19. What is the probability of rolling an odd number on a die?
20. If you have 5 apples and 3 oranges, what is the probability of picking an apple?

Medium Level Questions (21-40)

21. If you roll two dice, what is the probability of both showing odd numbers?
22. What is the probability of drawing two hearts in a row from a deck of cards without replacement?
23. If you flip a coin 4 times, what is the probability of getting exactly 2 heads?
24. You have 4 red balls and 6 blue balls. What is the probability of picking a blue ball first and then a red ball?
25. If you have a standard deck of cards, what is the probability of drawing a Queen and then a King without replacement?
26. What is the probability of rolling a sum of 9 with two dice?
27. If you draw a card from a deck and do not replace it, what is the probability of drawing two Aces in a row?
28. What is the probability of getting two tails when flipping a coin twice?
29. If you have a bag with 2 blue, 3 red, and 1 green ball, what is the probability of picking a red ball first and then a blue ball?
30. What is the probability of getting at least one 6 when rolling a die three times?
31. You have a box with 10 fruits (6 apples, 4 bananas). What is the probability of picking an apple and then a banana without replacement?
32. If you flip a coin 5 times, what is the probability of getting exactly 3 heads?
33. If you roll two dice, what is the probability of at least one die showing a 2?
34. What is the probability of drawing two Kings in a row from a standard deck of cards without replacement?
35. If you have a bag with 3 red and 2 green marbles, what is the probability of picking a green marble first and then a red marble?
36. What is the probability of drawing an Ace and then a Heart from a deck of cards without replacement?
37. If you flip a coin 3 times, what is the probability of getting two heads and one tail?
38. What is the probability of drawing a diamond and then a club from a deck of cards without replacement?
39. If you roll a die, what is the probability of rolling a number greater than 4?
40. You have 5 blue marbles and 7 yellow marbles. What is the probability of picking a yellow marble first and then a blue marble?

Hard Level Questions (41-60)

41. If you draw three cards from a deck without replacement, what is the probability of getting three aces?
42. What is the probability of rolling two dice and getting a total of 11?
43. If you flip a coin 6 times, what is the probability of getting 4 heads and 2 tails?
44. If a box contains 3 red, 5 blue, and 2 green balls, what is the probability of selecting a red ball first and then a blue ball?
45. What is the probability of drawing two hearts in a row from a deck, with replacement?
46. If you roll three dice, what is the probability of getting at least one 1?
47. If you draw a card from a deck, replace it, and then draw again, what is the probability of getting a heart both times?
48. What is the probability of getting 4 tails in 5 flips of a coin?
49. If you have a jar with 5 red, 4 blue, and 3 green marbles, what is the probability of picking a green marble, replacing it, and then picking a blue marble?
50. What is the probability of rolling two dice and both showing different numbers?
51. If you flip a coin 10 times, what is the probability of getting exactly 5 heads?
52. If you have a box with 10 items (4 of which are defective), what is the probability of picking a non-defective item and then a defective one without replacement?
53. What is the probability of picking a King, replacing it, and then picking a Queen from a deck of cards?
54. If you roll two dice, what is the probability of both showing even numbers?
55. If you have 7 cards (3 red, 4 black), what is the probability of drawing two black cards in a row without replacement?
56. If you flip a coin 8 times, what is the probability of getting at least 6 heads?
57. If you draw three cards from a deck with replacement, what is the probability of getting at least one heart?
58. If you have a box with 5 blue, 5 red, and 5 green balls, what is the probability of picking a red and a blue ball in succession with replacement?
59. If you roll a die twice, what is the probability that you get a 6 on the first roll and a number less than 4 on the second roll?
60. What is the probability of drawing two Aces in a row from a deck, with replacement?

Answers and Explanations

Easy Level Answers (1-20)

1. $$\frac{1}{2}$$
2. $$\frac{3}{6} = \frac{1}{2}$$
3. $$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$
4. $$\frac{2}{5}$$
5. $$\frac{6}{36} = \frac{1}{6}$$
6. $$\frac{13}{52} = \frac{1}{4}$$
7. $$\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$
8. $$\frac{4}{52} = \frac{1}{13}$$
9. $$\frac{1}{6}$$
10. $$1 – P(\text{no heads}) = 1 – \left(\frac{1}{2}\right)^3 = \frac{7}{8}$$
11. $$\frac{5}{10} = \frac{1}{2}$$
12. $$\frac{5}{6}$$
13. $$\frac{4}{52} = \frac{1}{13}$$
14. $$\frac{1}{2}$$
15. $$\frac{2}{5}$$
16. $$\frac{3}{6} = \frac{1}{2}$$
17. $$\frac{13}{52} = \frac{1}{4}$$
18. $$1 – P(\text{no tails}) = 1 – \left(\frac{1}{2}\right)^2 = \frac{3}{4}$$
19. $$\frac{3}{6} = \frac{1}{2}$$
20. $$\frac{5}{8}$$

Medium Level Answers (21-40)

21. $$\frac{3}{36} = \frac{1}{12}$$
22. $$\frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} = \frac{1}{17}$$
23. $${6 \choose 2} \left(\frac{1}{2}\right)^4 = 15 \times \frac{1}{16} = \frac{15}{16}$$
24. $$\frac{6}{10} \times \frac{4}{9} = \frac{24}{90} = \frac{4}{15}$$
25. $$\frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663}$$
26. $$\frac{4}{36} = \frac{1}{9}$$
27. $$\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$
28. $$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$$
29. $$\frac{3}{6} \times \frac{2}{5} = \frac{6}{30} = \frac{1}{5}$$
30. $$1 – P(\text{no 6s}) = 1 – \left(\frac{5}{6}\right)^3 = \frac{91}{216}$$
31. $$\frac{6}{10} \times \frac{4}{9} = \frac{24}{90} = \frac{4}{15}$$
32. $${10 \choose 4} \left(\frac{1}{2}\right)^{10} = 210 \times \frac{1}{1024} = \frac{105}{512}$$
33. $$1 – P(\text{no 2s}) = 1 – \left(\frac{5}{6}\right)^2 = \frac{7}{36}$$
34. $$\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$
35. $$\frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}$$
36. $$\frac{4}{52} \times \frac{13}{51} = \frac{52}{2652} = \frac{1}{51}$$
37. $${3 \choose 2} \left(\frac{1}{2}\right)^3 = 3 \times \frac{1}{8} = \frac{3}{8}$$
38. $$\frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} = \frac{1}{17}$$
39. $$\frac{2}{6} = \frac{1}{3}$$
40. $$\frac{5}{12} \times \frac{7}{11} = \frac{35}{132}$$

Hard Level Answers (41-60)

41. $$\frac{4}{52} \times \frac{3}{51} \times \frac{2}{50} = \frac{24}{132600} = \frac{1}{5525}$$
42. $$\frac{2}{36} = \frac{1}{18}$$
43. $${6 \choose 4} \left(\frac{1}{2}\right)^{6} = 15 \times \frac{1}{64} = \frac{15}{64}$$
44. $$\frac{3}{10} \times \frac{5}{9} = \frac{15}{90} = \frac{1}{6}$$
45. $$\frac{13}{52} \times \frac{12}{52} = \frac{156}{2704} = \frac{39}{676}$$
46. $$1 – P(\text{no 1s}) = 1 – \left(\frac{5}{6}\right)^3 = \frac{91}{216}$$
47. $$\frac{13}{52} \times \frac{13}{52} = \frac{169}{2704} = \frac{13}{208}$$
48. $${5 \choose 4} \left(\frac{1}{2}\right)^{5} = 5 \times \frac{1}{32} = \frac{5}{32}$$
49. $$\frac{3}{12} \times \frac{5}{12} = \frac{15}{144} = \frac{5}{48}$$
50. $$1 – P(\text{same numbers}) = 1 – \frac{6}{36} = \frac{5}{6}$$
51. $${10 \choose 5} \left(\frac{1}{2}\right)^{10} = 252 \times \frac{1}{1024} = \frac{63}{256}$$
52. $$\frac{6}{10} \times \frac{4}{9} = \frac{24}{90} = \frac{4}{15}$$
53. $$\frac{4}{52} \times \frac{4}{52} = \frac{16}{2704} = \frac{1}{169}$$
54. $$\frac{18}{36} = \frac{1}{2}$$
55. $$\frac{4}{7} \times \frac{3}{6} = \frac{12}{42} = \frac{2}{7}$$
56. $${8 \choose 6} \left(\frac{1}{2}\right)^{8} = 28 \times \frac{1}{256} = \frac{7}{64}$$
57. $$1 – P(\text{no hearts}) = 1 – \left(\frac{39}{52}\right)^3 = \frac{2391}{140608}$$

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