Complementary, sure and impossible events
An event E and 'not E' are complementary: P(E) + P(not E) = 1. A sure event has P = 1; an impossible event has P = 0.
Not E
The event not E (written Ē) happens exactly when E does not. If E is 'getting a 6', then not E is 'getting 1, 2, 3, 4 or 5'. E and not E are complementary events. Together they cover every outcome, so P(E) + P(not E) = 1.
From impossible to sure, with one die
| Type of event | Probability | Example with one die |
|---|---|---|
| Impossible event | 0 | Getting 8 |
| Event with one outcome | 1/6 | Getting a 4 |
| Even chance | 1/2 | Getting an even number |
| Likely event | 5/6 | Getting a number other than 6 |
| Sure (certain) event | 1 | Getting a number less than 7 |
Worked example 1: using the complement
The probability that a bus from Madurai reaches on time is 0.85. Find the probability that it does not reach on time.
- Let E = 'the bus reaches on time'. P(E) = 0.85.
- Formula: P(not E) = 1 − P(E).
- P(not E) = 1 − 0.85 = 0.15.
P(not on time) = 0.15.
Worked example 2: faulty bulbs
A box has 500 LED bulbs, of which 15 are faulty. One bulb is taken out at random. Find the probability that it is good, using the complement.
- Formula: P(E) = favourable outcomes ÷ total outcomes, and P(not E) = 1 − P(E).
- P(faulty) = 15/500 = 3/100.
- 'Good' and 'faulty' are complementary events.
- P(good) = 1 − 3/100 = 97/100 = 0.97.
- Check directly: good bulbs = 500 − 15 = 485, and 485/500 = 97/100.
P(good) = 97/100.
Which numbers can be a probability?
- Any number from 0 to 1, including 0 and 1: 0, 0.3, 2/3, 1.
- As a percentage, anything from 0% to 100%: 35% = 0.35.
- Never negative: −0.2 is not a probability.
- Never more than 1: 1.5, 5/4 and 120% are not probabilities.
- The probabilities of all the elementary events of an experiment add up to 1.
If a weather app says the chance of rain in Mumbai today is 70%, then the chance of no rain is 30%. You use the complement to decide whether to carry an umbrella.
1 marks: “If P(E) = 0.07, what is P(not E)?” P(not E) = 1 − 0.07 = 0.93.
1 marks: “Which of these cannot be a probability: 3/8, −0.6, 40%, 1?” −0.6, because a probability can never be negative.
2 marks: “A die is thrown once. Give one example each of a sure event and an impossible event, with their probabilities.” Sure: getting a number from 1 to 6, P = 1. Impossible: getting 7, P = 0.
E and not E share the whole, so they add to 1. Zero means never; one means always.