Probability · Ages 14–15
Certain, impossible and random events
All the outcomes: certain. None: impossible. Some: random.
How to do it
Method
All the outcomes: certain. None: impossible. Some: random
A die: "a 7" impossible, "less than 7" certain, "even" random
Another way
The unluckiest case
3 red, 4 blue: draw at least 4 + 1 = 5 to be sure of a red
In full
An event that must happen is certain; one that can't happen is impossible; one that may or may not happen is random. One way to tell: list every possible outcome and see how many the event takes in. A die shows 1 to 6: "a 7" takes in none of them, so it is impossible; "less than 7" takes in all 6, so it is certain; "an even number" takes in only 2, 4 and 6, so it is random. Another way: think of the unluckiest case. A bag holds 3 red and 4 blue balls; for a red ball to be certain, the unluckiest draw takes all 4 blue first, so at least 4 + 1 = 5 must be drawn; with 4 or fewer, "a red ball" is only random.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
For an ordinary die, "rolling a number greater than 3" is a certain event.
True or false?
Show the answer and working
Answer: False
A die shows 1 to 6. The event is certain only if all of 1 to 6 are greater than 3, that is, if 3 is less than 1.
Example 2
A bag holds 9 balls numbered 1 to 9, and one is drawn at random. For "the number drawn is greater than k" to be impossible, the least k can be is .
Show the answer and working
Answer: 9
The largest number is 9. When k is 9 or more, no number beats it; when k is less than 9, drawing 9 does.
Example 3
Which of these is a random event?
- A. Among 19 people, at least two born in the same month
- B. Tossing a coin and getting heads
- C. A year having 14 months
- D. A stone thrown up into the air coming down
Choose the right answer
Show the answer and working
Answer: B. Tossing a coin and getting heads
A coin may land heads or tails; the outcome can't be known beforehand, so it is random.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.An ordinary die is rolled once. Which of these is a certain event?
- A. Rolling a 8
- B. Rolling an odd number
- C. Rolling a number less than 9
- D. Rolling a number less than 2
Choose the right answer
Answer:
2.An ordinary die is rolled once. Which of these is an impossible event?
- A. Rolling a number less than 7
- B. Rolling a 3
- C. Rolling a 7
- D. Rolling an even number
Choose the right answer
Answer:
3.A bag holds 1 red balls and 4 green ones, and nothing else. One is drawn at random; "a red ball" is a random event.
True or false?
Answer:
4.A bag holds 18 balls numbered 1 to 18, and one is drawn at random. For "the number drawn is greater than k" to be impossible, the least k can be is .
Answer:
5.Which of these is a random event?
- A. A stone thrown up into the air coming down
- B. Tossing a coin and getting heads
- C. A year having 14 months
- D. Among 14 people, at least two born in the same month
Choose the right answer
Answer:
All the answers
- 1. C. Rolling a number less than 9
- 2. C. Rolling a 7
- 3. True
- 4. 18
- 5. B. Tossing a coin and getting heads
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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