Statistics and Probability
Probability
Work out the probability of a simple event and of it not happening.
Roll an ordinary die and six things can happen, each as likely as the others. Two of them are a 5 or a 6. So the probability of rolling more than 4 is two out of six. That is the whole of theoretical probability: count the outcomes you want, count all the equally likely outcomes there are, and write the first over the second.
The rule
P(E) = number of wanted outcomes ÷ total number of equally likely outcomes. It is never below 0 and never above 1, because you can never want fewer than none of the outcomes or more than all of them. An impossible event has probability 0; a certain one has probability 1.
The words "equally likely" are doing real work. The rule counts outcomes, so it only holds when the outcomes really are interchangeable — a fair die, a well-shuffled pack, a bag you cannot see into. "It rains tomorrow or it does not" is two outcomes and nothing like a half.
Worked example
A fair die is rolled. What is the probability of an even number?
The outcomes are 1, 2, 3, 4, 5 and 6 — six of them, all equally likely.
Always list the whole set first. Most mistakes in this topic are a miscounted bottom line, not a miscounted top one.
The event that does not happen
Everything either happens or it does not, so those two probabilities add to 1. If the probability of rain is 0.3, the probability of no rain is 0.7. When counting what you want is fiddly, count what you do not want and subtract — it is often far quicker.
Try it together
A bag holds 4 green, 6 yellow and 10 white counters. One counter is taken out without looking.
Total first, then the count of what you want, then the fraction.
1.The bag holds 4 green, 6 yellow and 10 white counters. How many counters are there altogether?
Have a go
Have a go on your own: a bag holds 5 red and 7 green marbles. One is taken out without looking. What is the probability that it is green? Give your answer as a fraction.
Hint: Count every marble in the bag for the bottom of the fraction.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.