What is probability?

How likely is it?

A bag holds $3$ red, $2$ blue and $5$ green marbles. You take one marble without looking. Which colour do you expect? Green is the most likely, because there are more green marbles. A probability puts a number on how likely.

A probability is always a number from $0$ to $1$.

• $0$ means that it cannot happen.
• $1$ means that it will happen.
• $0.5$ means an even chance.

A probability is never less than $0$ or more than $1$.

Equally likely outcomes

When every outcome has the same chance, we find the probability by counting.

You should find that the three parts of the bar always fill it exactly, from 0 to 1. Add green marbles, and the red and blue parts shrink, though their counts stay the same.

Try it in Python

Listing the outcomes

One die has six outcomes, $1, 2, 3, 4, 5, 6$, all equally likely. To find a probability, list the outcomes that we want, and count them.

The chance that it does not happen

Either an event happens or it does not, so the two probabilities add up to $1$.

The chance of a six is $\dfrac{1}{6}$, so the chance of not a six is $1 - \dfrac{1}{6} = \dfrac{5}{6}$. This is often the easy way to a hard probability.

Practise with new numbers

Common mistakes

• A probability above 1, or below 0. It cannot be. Check the count again.
• Dividing by the wrong total. Divide by all the outcomes, not by the ones you do not want.
• Counting outcomes that are not equally likely. Win or lose is two outcomes, but they need not be equally likely.
• Forgetting the short cut. For "not", take the chance away from 1.

Remember these

• A probability is a number from $0$ to $1$.
• With equally likely outcomes: the wanted outcomes divided by all the outcomes.
• $P(\text{not } A) = 1 - P(A)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A bag has $4$ red and $6$ blue marbles. What is $P(\text{blue})$?

2. The chance of rain is $0.35$. What is the chance of no rain?

3. A die is rolled. What is the probability of a number less than 3?

4. Which of $1.2$, $-0.4$ and $0.85$ could be a probability?