Happening, and not happening
The weather forecast says that the probability of rain tomorrow is . Draw it as a bar cut into 5 equal parts, and color 1 part for rain.
1 of the 5 equal parts is colored: the probability of rain is .
The complement fills the rest
Tomorrow it will either rain or stay dry, and it cannot do both. So the other 4 parts of the bar are the probability that it does not rain: . The event "it does not rain" is called the complement of the event "it rains".
An event and its complement together fill the whole bar, and the whole bar is 1, certain: . So the probability that something does not happen is 1 minus the probability that it does: .
The same bar with every part named. 1 part is rain and the other 4 parts are dry: is the whole bar, 1.
On the probability scale, rain takes the first and dry takes the other . Together they reach 1, certain.
Sometimes the complement is quicker
What is the probability of not rolling a 6 on a fair dice? Five of the six equally likely outcomes are not a 6, so the probability is . Or start from the 6: its probability is , so the probability of not rolling a 6 is .
Decimals work the same way. If the probability that a bus is late is 0.25, the probability that it is not late is 1 − 0.25 = 0.75.
The 6 outcomes of a dice, with the 5 that are not a 6 colored: , which is .
Everything that is not the event
The complement is every outcome that is not the event, not just one opposite word. A bus can be early, on time or late. The complement of "late" is "early or on time", so the probability that the bus is not late is not the same as the probability that it is on time.
The complement is not the fraction turned over, either. The complement of is , not : is 5, and no probability is more than 1. And it is not the same fraction again: is the chance of rain, and the chance of no rain is the rest of the bar.
Worked example: A City Bus That Can Be Early, On Time or Late
Question The Route 9 bus takes Mia to school. On any school day, the probability that it is late is 0.25 and the probability that it is early is 0.1. Otherwise it is on time. (a) What is the probability that the bus is not late? (b) What is the probability that the bus is on time?
1.Draw one bar for all the school days, with probability 1. Mark 0.1 at one end for early and 0.25 at the other end for late.
The whole bar is all the school days, probability 1: early 0.1 and late 0.25. 2.(a) Not late is the complement of late, so its probability is 1 − 0.25 = 0.75.
(a) Not late is the complement of late: 1 − 0.25 = 0.75. 3.The days that are not late are the early days and the on-time days together.
The days that are not late are the early days and the on-time days. 4.(b) Take away the early days: 0.75 − 0.1 = 0.65. Check: 0.1 + 0.65 + 0.25 = 1.
(b) On time is 0.75 − 0.1 = 0.65. Check: 0.1 + 0.65 + 0.25 = 1.
Answer: (a) 0.75; (b) 0.65
Common mistakes
- Answering part (b) with 0.75. The days that are not late include the early days, and an early bus is not on time, so take away the 0.1 as well.
- Giving 0.25 + 0.1 = 0.35 as the probability that the bus is on time. That is the probability that it is early or late, the opposite of on time, so on time is 1 − 0.35 = 0.65.
Worked example: A Tombola at the School Fair, and the Chance That a Ticket Wins Nothing
Question A tombola drum at the school fair holds 200 folded tickets. 30 of them win a prize and the rest win nothing. Each person takes one ticket without looking, and no ticket goes back. (a) What is the probability that the first ticket taken wins nothing? (b) By noon, 50 tickets have been taken, and 10 of them won prizes. What is the probability that the next ticket taken wins nothing?
1.(a) 200 − 30 = 170 tickets win nothing. The probability is 170200 = 1720. Check with the complement: 1 − 30200 = 170200.
(a) 200 − 30 = 170 tickets win nothing: 170200 = 1720. 2.At noon, 200 − 50 = 150 tickets are left in the drum.
By noon 50 tickets are gone: 10 that won and 40 that won nothing. 150 are left. 3.10 prizes have gone, so 30 − 10 = 20 winning tickets are left. The other 150 − 20 = 130 tickets win nothing.
30 − 10 = 20 winners are left, so 150 − 20 = 130 tickets win nothing. 4.(b) The probability that the next ticket wins nothing is 130150 = 1315.
(b) 130150 = 1315.
Answer: (a) 1720; (b) 1315
Common mistakes
- Keeping 200 as the total in part (b). The 50 tickets taken are not in the drum any more, so the next ticket comes from 150.
- Taking all 50 tickets away from the 170 that win nothing. Only 40 of the 50 won nothing, so 170 − 40 = 130 are left, the same as 150 − 20.