Mathematics Year 7: Probability of an event with several outcomes
Mathematics, Year 7 (Portugal): Given a probability model, the probability of an event made of several outcomes is the sum of the probabilities of those outcomes, even when the outcomes are not equally likely.
Add the outcomes that make up the event
An outcome is one single thing that can come up: one face of a die, one colour on a spinner. Often you want the probability of something made of several outcomes: “a multiple of 3” takes in the 3 and the 6. The probability of that event is the sum of the probabilities of the outcomes that make it up. With a fair die: P(multiple of 3) = P(3) + P(6) = 1/6 + 1/6 = 2/6 = 1/3. (Here “event” is used informally.)
One model answers many questions
Instead of running an experiment for every event, one probability model is enough: the list of all the outcomes with the probability of each, and all of them together must give 1 (or 100%). With the model in hand, any event is found by adding. You do not need equal chances: the outcomes in the model can have different probabilities, and the sum still works. It is the same idea as in earlier years, now used with a model someone gives you.
A spinner with five outcomes
A spinner has five outcomes, with this model: A = 6/20, B = 5/20, C = 4/20, D = 3/20, E = 2/20. Check: 6 + 5 + 4 + 3 + 2 = 20, and 20/20 = 1. What is the probability of A, B or C? Step 1: P(A or B or C) = P(A) + P(B) + P(C). Step 2: 6/20 + 5/20 + 4/20 = 15/20. Step 3: 15/20 = 3/4 = 75%. And D or E? 3/20 + 2/20 = 5/20 = 1/4 = 25%. Check: 75% + 25% = 100%, because together they are all the outcomes.
Two outcomes out of five is not 40%
On the same spinner, what is the probability of A or B? Many people count the outcomes: “2 out of 5, so 2/5 = 40%”. But that only works when all outcomes are equally likely, and here they are not: A has 30% and B has 25%. The right answer is P(A) + P(B) = 6/20 + 5/20 = 11/20 = 55%. With a fair die, counting and adding give the same, because every face has 1/6, but adding is the method that always works.
Forecasts and raffles
Weather forecasts use models like this. If the model says sun 50%, clouds 30%, rain 20%, the probability of no rain is that of sun or clouds: 50% + 30% = 80%. In a raffle with 100 tickets where 5 win a book, 10 a pen and 15 a sticker, the probability of winning some prize is 5% + 10% + 15% = 30%. Whenever you are asked “what is the probability of this or that?”, add the probabilities of the outcomes.
Test what you learned
With a fair die, “a multiple of 3” takes in the outcomes 3 and 6. What is its probability?
- a) 1/6
- b) 1/3, because 1/6 + 1/6 = 2/6
- c) 1/36
- d) 9/6
Correct answer: b) 1/3, because 1/6 + 1/6 = 2/6 — Right: P(multiple of 3) = P(3) + P(6) = 1/6 + 1/6 = 2/6 = 1/3. The probability of the event is the sum of the probabilities of its outcomes.
A spinner has the model A = 6/20, B = 5/20, C = 4/20, D = 3/20, E = 2/20. What is the probability of getting A, B or C?
- a) 3/5 = 60%, three outcomes out of five
- b) About 1.5%, multiplying 6/20 × 5/20 × 4/20
- c) 3/4 = 75%
- d) 1/4 = 25%, the mean of the three
Correct answer: c) 3/4 = 75% — P(A) + P(B) + P(C) = 6/20 + 5/20 + 4/20 = 15/20 = 3/4 = 75%. Adding the probabilities of the outcomes always works.
On the same spinner (A = 30%, B = 25%, C = 20%, D = 15%, E = 10%), what is the probability of getting A or B?
- a) 55%, because 30% + 25%
- b) 40%, two outcomes out of five
- c) 30%, the more likely of the two
- d) 27.5%, the mean of 30% and 25%
Correct answer: a) 55%, because 30% + 25% — Right: P(A) + P(B) = 30% + 25% = 55% (11/20). The outcomes are not equally likely, so you add their probabilities.
Keep exploring
Other languages
Loading MyLeoNes™…