Year 11 Maths: Probability of Compound Events

Understanding Probability of Compound Events

What is Probability?

Probability is the chance of something happening. We express it as a number between 0 and 1. A probability of 0 means an event will not happen, while a probability of 1 means it will definitely happen.

What are Compound Events?

Compound events are situations where two or more events happen at the same time or in sequence. We can combine these events in two main ways:

1. Independent Events: These are events where the outcome of one event does not affect the outcome of another. For example, flipping a coin and rolling a die. The coin flip does not influence what number comes up on the die.
2. Dependent Events: These are events where the outcome of one event affects the outcome of another. For example, drawing two cards from a deck without replacing the first card. The first card changes the probability of the second card.

Key Rules for Probability of Compound Events

1. Independent Events

For two independent events A and B, the probability of both A and B occurring is given by:

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

Example: If the probability of flipping heads on a coin is $$\frac{1}{2}$$ and the probability of rolling a 3 on a die is $$\frac{1}{6}$$ , the probability of both happening is:

$$P(\text{Heads and 3}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

2. Dependent Events

For two dependent events A and B, the probability of both A and B occurring is:

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

Where $$P(B | A)$$ is the probability of event B occurring given that event A has occurred.

Example: If you draw a card from a deck and it’s an Ace, the probability of drawing another Ace without replacing the first card is:

$$P(\text{first Ace}) = \frac{4}{52}$$

$$P(\text{second Ace | first Ace}) = \frac{3}{51}$$

So,

$$P(\text{Ace and Ace}) = P(\text{first Ace}) \times P(\text{second Ace | first Ace}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$

Tips and Tricks

• Draw a Diagram: Sometimes, drawing a Venn diagram or a tree diagram can help visualise the events.
• Organise Your Work: Write down the probabilities you know and how they relate to each other.
• Practice: Use lots of examples to get comfortable with both independent and dependent events.
• Check Your Work: Always review your calculations to catch any mistakes.

Practice Questions

Easy Level (20 Questions)

1. What is the probability of rolling a 4 on a six-sided die?
2. If you flip a coin, what is the probability it lands on tails?
3. What is the probability of drawing a heart from a standard deck of cards?
4. If you roll two dice, what is the probability of both showing a number greater than 2?
5. What is the probability of drawing a red card from a deck of cards?
6. If you flip a coin twice, what is the probability of getting at least one head?
7. What is the probability of rolling a number less than 5 on a six-sided die?
8. If you draw a card from a full deck, what is the probability it is a King?
9. What is the probability of rolling a total of 7 with two dice?
10. If you flip three coins, what is the probability of getting exactly two heads?
11. What is the probability of a 7 on the first roll of a pair of dice?
12. What is the probability of choosing a blue marble from a bag containing 5 blue marbles and 3 red marbles?
13. What is the probability of getting a prime number when rolling a die?
14. If you roll one die, what is the probability of rolling an even number?
15. What is the probability of drawing a Queen from a deck of cards?
16. If you flip a coin and roll a die, what is the probability of getting tails and a 6?
17. What is the probability of rolling two dice and getting doubles?
18. What is the probability of not rolling a 1 on a six-sided die?
19. If you draw one card from a deck, what is the probability it is not a heart?
20. What is the probability of flipping a coin and getting two heads in two flips?

Medium Level (20 Questions)

1. If the probability of raining tomorrow is $$\frac{2}{5}$$ , what is the probability it will not rain?
2. What is the probability of rolling a sum of 8 with two dice?
3. There are 10 marbles (6 red, 4 blue). If you draw one, what is the probability it is blue?
4. What is the probability of drawing two aces in a row from a deck of cards without replacement?
5. If you flip a coin and roll a die, what is the probability of getting heads and an odd number?
6. What is the probability of drawing a club or a heart from a deck of cards?
7. If you roll three dice, what is the probability of getting at least one 6?
8. What is the probability of drawing a face card from a standard deck of cards?
9. If you flip a coin three times, what is the probability of getting at least one tail?
10. What is the probability of drawing two cards from a deck and both are red?
11. If you roll a pair of dice, what is the probability of getting a sum of 3?
12. What is the probability of flipping a coin twice and getting HT?
13. If you have a bag with 3 blue and 2 green marbles, what is the probability of drawing a blue marble?
14. What is the probability of drawing a King or a 10 from a deck of cards?
15. If you randomly select a day of the week, what is the probability it is a weekend?
16. What is the probability of rolling a number greater than 4 on a six-sided die?
17. If you draw a card from a deck, what is the probability it is a face card given it is a red card?
18. What is the probability of flipping two coins and getting at least one head?
19. What is the probability of rolling a number that is not a 3 on a die?
20. If you have a standard deck of cards, what is the probability of drawing a red card first and then a black card?

Hard Level (20 Questions)

1. If you draw two cards from a deck without replacement, what is the probability both are spades?
2. What is the probability of rolling a total of 9 with two dice?
3. If the probability of an event A is $$\frac{1}{4}$$ and event B is $$\frac{1}{3}$$ , what is the probability of both events occurring if they are independent?
4. What is the probability of getting two heads and one tail in three flips of a coin?
5. If you have 5 different books, what is the probability of picking a specific book from the group?
6. What is the probability of not drawing a King in two successive draws from a deck without replacement?
7. If the probability of event A occurring is $$\frac{5}{8}$$ and event B is $$\frac{3}{4}$$ , what is the probability of A and B occurring together if they are dependent?
8. What is the probability of rolling two dice and getting a sum of at least 10?
9. If you flip three coins, what is the probability of getting exactly one head?
10. If you draw three cards from a deck without replacement, what is the probability they are all hearts?
11. What is the probability of rolling a number less than 3 or greater than 4 on a die?
12. If there are 3 red and 2 green marbles, what is the probability of drawing a red marble, replacing it, and drawing a green marble?
13. What is the probability of drawing a diamond or a face card from a deck of cards?
14. If the probability of event A is $$\frac{2}{5}$$ , what is the probability of A not occurring?
15. What is the probability of rolling two dice and getting a sum of 6 or less?
16. If you have a box containing 4 red, 3 green, and 2 blue balls, what is the probability of drawing a blue ball followed by a red ball without replacement?
17. If the probability of a student passing a test is $$\frac{3}{5}$$ , what is the probability of 3 students passing if they are independent?
18. What is the probability of drawing a card that is either a Queen or a Club?
19. If you have 2 coins and a die, what is the probability of getting at least one tail and rolling a number less than 3?
20. If you draw two cards from a standard deck with replacement, what is the probability that both cards are black?

Answers to Questions

Answers for Easy Level

1. $$\frac{1}{6}$$
2. $$\frac{1}{2}$$
3. $$\frac{13}{52} = \frac{1}{4}$$
4. $$\frac{4}{36} = \frac{1}{9}$$
5. $$\frac{26}{52} = \frac{1}{2}$$
6. $$\frac{3}{4}$$
7. $$\frac{4}{6} = \frac{2}{3}$$
8. $$\frac{4}{52} = \frac{1}{13}$$
9. $$\frac{6}{36} = \frac{1}{6}$$
10. $$\frac{3}{8}$$
11. $$\frac{1}{6}$$
12. $$\frac{4}{7}$$
13. $$\frac{3}{6} = \frac{1}{2}$$
14. $$\frac{1}{2}$$
15. $$\frac{3}{6} = \frac{1}{2}$$
16. $$\frac{1}{13}$$
17. $$\frac{1}{36}$$
18. $$\frac{1}{4}$$
19. $$\frac{5}{6}$$
20. $$\frac{3}{8}$$

Answers for Medium Level

1. $$\frac{3}{5}$$
2. $$\frac{5}{36}$$
3. $$\frac{2}{5}$$
4. $$\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$
5. $$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$
6. $$\frac{26}{52} + \frac{13}{52} – \frac{13}{52} = \frac{26}{52} = \frac{1}{2}$$
7. $$1 – P(\text{no 6}) = 1 – \left(\frac{5}{6}\right)^3 = 1 – \frac{125}{216} = \frac{91}{216}$$
8. $$\frac{12}{52} = \frac{3}{13}$$
9. $$1 – \left(\frac{1}{2}\right)^3 = 1 – \frac{1}{8} = \frac{7}{8}$$
10. $$\frac{26}{52} \times \frac{25}{51} = \frac{650}{2652} = \frac{25}{102}$$
11. $$\frac{2}{36} = \frac{1}{18}$$
12. $$\frac{3}{5}$$
13. $$\frac{4}{13}$$
14. $$\frac{2}{7}$$
15. $$\frac{2}{7}$$
16. $$\frac{4}{5} \times \frac{3}{5} = \frac{12}{25}$$
17. $$1 – P(\text{not a face}) = 1 – \frac{40}{52} = \frac{12}{52} = \frac{3}{13}$$
18. $$\frac{1}{2}$$
19. $$\frac{5}{6}$$
20. $$\frac{1}{2}$$

Answers for Hard Level

1. $$\frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} = \frac{1}{17}$$
2. $$\frac{4}{36} = \frac{1}{9}$$
3. $$\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$$
4. $$\frac{3}{8}$$
5. $$\frac{1}{5}$$
6. $$1 – \frac{4}{52} \times \frac{3}{51} = 1 – \frac{12}{2652} = \frac{2640}{2652}$$
7. $$\frac{5}{8} \times \frac{1}{4} = \frac{5}{32}$$
8. $$\frac{5}{36}$$
9. $$\frac{3}{8}$$
10. $$\frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} = \frac{1716}{132600} = \frac{11}{850}$$
11. $$\frac{2}{6} = \frac{1}{3}$$
12. $$\frac{3}{9} \times \frac{2}{8} = \frac{6}{72} = \frac{1}{12}$$
13. $$\frac{13}{52} + \frac{12}{52} – \frac{3}{52} = \frac{22}{52} = \frac{11}{26}$$
14. $$\frac{3}{5}$$
15. $$\left(1 – \frac{5}{6} \times \frac{2}{6}\right) = \frac{19}{36}$$
16. $$\frac{2}{9} \times \frac{4}{8} = \frac{8}{72} = \frac{1}{9}$$
17. $$\left(\frac{3}{5}\right)^3 = \frac{27}{125}$$
18. $$\frac{4}{52} + \frac{12}{52} – \frac{4}{52} = \frac{12}{52} = \frac{3}{13}$$
19. $$\frac{1}{2}$$
20. $$\left(\frac{1}{2} \times \frac{1}{2}\right) = \frac{1}{4}$$

This structured approach to discussing the probability of compound events will help you grasp the concepts and prepare for your assessments. Keep practicing, and don’t hesitate to ask questions if you need clarification!