Year 9 Maths: Probability of Mutually Exclusive Events and Overlapping Events

Introduction to Probability

Probability is the chance of something happening. We often express this as a number between 0 and 1. A probability of 0 means it will never happen, while a probability of 1 means it will definitely happen.

In this lesson, we will focus on two types of events: mutually exclusive events and overlapping events.

What Are Mutually Exclusive Events?

Mutually exclusive events are events that cannot happen at the same time. If one event happens, the other cannot.

Example of Mutually Exclusive Events

Imagine you have a bag of fruit with an apple, a banana, and an orange. If you pick one fruit, you can either pick an apple or a banana or an orange. You cannot pick both an apple and a banana at the same time.

• Events:
  • Picking an apple (A)
  • Picking a banana (B)

Since you can’t pick both, events A and B are mutually exclusive.

Key Rule for Mutually Exclusive Events

If two events A and B are mutually exclusive, the probability of either event A or event B occurring is:

$$P(A \text{ or } B) = P(A) + P(B)$$

What Are Overlapping Events?

Overlapping events are events that can happen at the same time. This means that one event can happen while another event is also happening.

Example of Overlapping Events

Let’s say you are rolling a die. The events of rolling an even number and rolling a number greater than 3 are overlapping.

• Events:
  • Rolling an even number (2, 4, 6)
  • Rolling a number greater than 3 (4, 5, 6)

Here, you can roll a 4 or a 6, which are counted in both events.

Key Rule for Overlapping Events

If two events A and B are overlapping, the probability of either A or B occurring is:

$$P(A \text{ or } B) = P(A) + P(B) – P(A \text{ and } B)$$

The last part, $$P(A \text{ and } B)$$ , is subtracted because we have counted those outcomes twice.

Tips and Tricks

• Draw a Venn diagram: This can help you visualise overlapping events.
• Use a probability tree: This helps to see all possible outcomes.
• Remember the rules: Always check if events are mutually exclusive or overlapping before calculating probabilities.

Questions

Easy Level Questions

1. If you flip a coin, what is the probability of getting heads?
2. A bag contains a red ball and a green ball. What is the probability of picking a red ball?
3. You roll a die. What is the probability of rolling a 1 or a 2?
4. If it’s raining, what is the probability that it is not sunny?
5. In a class of 30 students, 15 are boys. What is the probability of picking a boy?
6. A dice shows a number less than 4. What are the possible outcomes?
7. If you pick a day randomly, what is the probability that it is a Monday?
8. You have a box of chocolates, and there is one orange one. What is the chance of picking an orange chocolate?
9. When rolling a die, what is the chance of rolling an even number?
10. If I pick a fruit from a basket containing 2 apples and 3 bananas, what is the probability of picking a banana?

Medium Level Questions

11. What is the probability of rolling a number less than 5 on a 6-sided die?
12. In a class, 12 students play football and 8 students play basketball. What is the probability that a student plays football or basketball if no one plays both?
13. You draw a card from a standard deck of 52 cards. What is the probability of drawing a heart?
14. What is the probability of rolling a number greater than 4 on a die?
15. If you spin a spinner that is divided into 4 equal parts (A, B, C, D), what is the probability of landing on A or B?
16. A box has 5 red and 5 blue marbles. What is the probability of picking a red or blue marble?
17. In a bag with 4 apples and 6 oranges, what is the probability of picking either an apple or an orange?
18. A student has a 50% chance of passing maths and a 30% chance of passing science. What is the probability of passing either subject if they cannot pass both?
19. If you roll two dice, what is the probability of rolling a total of 7?
20. In a pack of cards, what is the probability of drawing a queen or a king?

Hard Level Questions

21. A bag contains 2 red, 3 blue, and 5 green marbles. What is the probability of picking either a red or blue marble?
22. If two dice are rolled, what is the probability of getting at least one 6?
23. In a standard deck of cards, what is the probability of drawing a heart or a face card?
24. You flip two coins. What is the probability of getting at least one head?
25. If you draw a card from a deck of cards and then another card without replacing the first, what is the probability of drawing two hearts?
26. In a group of 50 students, 20 play football, 15 play basketball, and 5 play both. What is the probability that a student plays either football or basketball?
27. You randomly select a month. What is the probability that it has 31 days?
28. A spinner is divided into 8 equal sections. What is the probability of landing on a number greater than 4 or an even number?
29. In a class of 40 students, 10 study French, 15 study Spanish, and 5 study both. What is the probability a student studies either language?
30. If the probability of passing a test is 0.6 and the probability of passing the exam is 0.7, what is the probability of passing at least one if they cannot pass both?

Answers

Easy Level Answers

1. $$P(\text{Heads}) = 0.5$$
2. $$P(\text{Red Ball}) = 0.5$$
3. $$P(1 \text{ or } 2) = \frac{2}{6} = \frac{1}{3}$$
4. $$P(\text{Not Sunny}) = 1 – P(\text{Raining})$$
5. $$P(\text{Boy}) = \frac{15}{30} = 0.5$$
6. Possible outcomes: 1, 2, 3.
7. $$P(\text{Monday}) = \frac{1}{7}$$
8. $$P(\text{Orange Chocolate}) = \frac{1}{\text{Total Chocolates}}$$
9. $$P(\text{Even Number}) = \frac{3}{6} = 0.5$$
10. $$P(\text{Banana}) = \frac{3}{5} = 0.6$$

Medium Level Answers

11. $$P(\text{Less than 5}) = \frac{4}{6} = \frac{2}{3}$$
12. $$P(\text{Football or Basketball}) = \frac{20}{30} = \frac{2}{3}$$
13. $$P(\text{Heart}) = \frac{13}{52} = \frac{1}{4}$$
14. $$P(\text{Greater than 4}) = \frac{2}{6} = \frac{1}{3}$$
15. $$P(A \text{ or } B) = \frac{2}{4} = \frac{1}{2}$$
16. $$P(\text{Red}) = \frac{5}{10} = 0.5$$
17. $$P(\text{Apple or Orange}) = 1$$
18. $$P(\text{Pass}) = 0.5 + 0.3 – 0 = 0.8$$
19. $$P(\text{Total 7}) = \frac{6}{36} = \frac{1}{6}$$
20. $$P(\text{Queen or King}) = \frac{8}{52} = \frac{2}{13}$$

Hard Level Answers

21. $$P(\text{Red or Blue}) = \frac{5}{10} = 0.5$$
22. $$P(\text{At least one 6}) = \frac{11}{36}$$
23. $$P(\text{Heart or Face}) = \frac{16}{52} = \frac{4}{13}$$
24. $$P(\text{At least one head}) = \frac{3}{4}$$
25. $$P(\text{Two Hearts}) = \frac{13}{52} \times \frac{12}{51}$$
26. $$P(\text{Football or Basketball}) = \frac{30}{50} = \frac{3}{5}$$
27. $$P(\text{31 Days}) = \frac{7}{12}$$
28. $$P(\text{>4 or Even}) = \frac{6}{8} = \frac{3}{4}$$
29. $$P(\text{Either Language}) = \frac{20}{40} = \frac{1}{2}$$
30. $$P(\text{Pass at least one}) = 0.6 + 0.7 – (0.6 \times 0.7) = 0.82$$

This structured approach to probability should help you understand the concepts of mutually exclusive and overlapping events. Remember, practice makes perfect!