I throw a pair of 6-sided dice. Write down an event that has a probability of 0 and an outcome that has a probability of 1.
Step-by-step solution
Idea: An impossible event has no favourable outcomes (P = 036 = 0). A certain event contains every outcome in the sample space (P = 3636 = 1). No single pair of numbers is certain (each has probability 136), so the probability-1 answer has to be an event that includes all 36 outcomes.
- The sum of two dice is at least 1 + 1 = 2 and at most 6 + 6 = 12.½ mark
- Probability 0: “The sum is 13.” No outcome gives 13, so the number of favourable outcomes is 0 and P = 036 = 0. (Other answers: “the sum is 1”, “a die shows 7”.)½ mark
- Probability 1: “The sum is between 2 and 12 (inclusive).” All 36 outcomes give such a sum, so P = 3636 = 1. (Other answers: “each die shows a number less than 7”, “the sum is less than 13”.)½ mark
- Note: a single outcome such as (3, 4) has probability 136, not 1. So the “outcome” with probability 1 must be an event made of all 36 outcomes, i.e. the whole sample space.½ mark
Check: Count: no pair adds to 13, and every one of the 36 pairs adds to a number from 2 to 12 ✓.
Answer to write in the exam
n(S) = 6 × 6 = 36; possible sums 2 to 12
Event A: sum = 13; n(A) = 0; P(A) = 036 = 0
Event B: sum between 2 and 12; n(B) = 36; P(B) = 3636 = 1
∴ Probability 0: the sum is 13. Probability 1: the sum is between 2 and 12.
Common mistakes that cost marks
- Giving “getting a double six” as the probability-0 event. It is possible (probability 136).
- Giving one particular pair such as (6, 6) as the probability-1 outcome. Any single pair has probability 136; only an event covering all outcomes is certain.
- Giving “the sum is 12” as impossible. 6 + 6 = 12 can happen.
How this can come in the exam
Two dice are thrown together. The probability that the product of the two numbers is 0 is
- 0
- 136
- 16
- 1
Show answer
(A) 0
Neither die shows 0, so the product is never 0: impossible, probability 0.
Assertion (A): When two dice are thrown, the probability that the sum is less than 13 is 1.
Reason (R): The probability of an event that includes every outcome of the sample space is 1.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Every sum is at most 12, so the event includes all 36 outcomes; R is exactly why A is true.
Try one yourself
A single die is thrown. Write an event with probability 0 and an event with probability 1.
Show answer
Probability 0: getting 8 (or getting 0). Probability 1: getting a number from 1 to 6 (or a number less than 7).
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