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Probability step by step

When all outcomes are equally likely, the probability is the number of favourable outcomes divided by the number of possible outcomes. Rolling a five with a dice has the probability 1/6, about 17%. A probability is always between 0 and 1. With two steps, we multiply along the path in a tree diagram.

Years 6–9 and upper secondary · Updated

Watch Fox work it out on the board

Fox works one step at a time. Press play.

Favourable divided by possible

P(event) = number of favourable outcomes ÷ number of possible outcomes

A dice has six equally likely outcomes. A five is 1 favourable outcome out of 6, so P(five) = 1/6.

EventFractionDecimalPercentage
Heads with a coin1/20.550%
A five with a dice1/6≈ 0.17≈ 17%
An even number with a dice3/6 = 1/20.550%
A seven with a dice0/6 = 000%, impossible
1 to 6 with a dice6/6 = 11100%, certain

The complementary event

Either something happens or it does not. Together that makes 1.

P(not A) = 1 − P(A)

The chance of not rolling a six is 1 − 1/6 = 5/6.

Two steps: tree diagrams

We toss a coin and roll a dice. Heads comes up half of the time, and a six in a sixth of those times. Along a path in the tree, we multiply.

P(heads and six) = 1/2 · 1/6 = 1/12 ≈ 0.083 ≈ 8.3%

If several paths give what we are looking for, we add the probabilities of those paths.

Example

Example 1: a marble from a bag

A bag has 3 red, 5 blue and 2 yellow marbles. We want to find P(blue).

  1. Possible outcomes: every marble has the same chance.

    3 + 5 + 2 = 10 possible outcomes

  2. Favourable outcomes: the blue marbles.

    5 favourable outcomes

  3. We divide favourable by possible.

    P(blue) = 510\frac{5}{10}

  4. We simplify and write it as a decimal and a percentage.

    510=12=0\frac{5}{10} = \frac{1}{2} = 0.5 = 50%

Half of the marbles are blue, so the chance should be a half. That is correct.

Example 2: two marbles without replacement

The bag has 3 red and 2 blue marbles. We draw two without replacement. We want to find P(both red).

  1. The first draw: 3 red out of 5.

    P(red first) = 35\frac{3}{5}

  2. Now 2 red out of 4 are left.

    P(red again) = 24\frac{2}{4}

  3. We multiply along the path red, red.

    35⋅24=620\frac{3}{5} \cdot \frac{2}{4} = \frac{6}{20}

  4. We simplify and write it as a percentage.

    620=310=0\frac{6}{20} = \frac{3}{10} = 0.3 = 30%

There are 5 · 4 = 20 ordered pairs. Two reds can be drawn in 3 · 2 = 6 ways. 6 out of 20 is 30%.

Pitfalls

  • Dividing by the other outcomes instead of all of them, so that 5 blue out of 10 marbles becomes 5/5.

    The denominator is all the possible outcomes, including the favourable ones.

  • Thinking that two coins give three equally likely outcomes and answering 1/3 for one of each.

    The outcomes are heads-heads, heads-tails, tails-heads and tails-tails. One of each: 2/4 = 1/2.

  • Adding along a path in the tree, so that heads and a six becomes 1/2 + 1/6.

    Along a path we multiply. We only add between different paths.

Practise on your own

0 of 4 correct

  1. You roll a dice. What is the probability of getting more than 4? Answer with a fraction.

  2. A bag has 4 green and 6 yellow marbles. What is the probability that a marble you take is not green?

  3. You toss a coin twice. What is the probability of getting heads both times?

  4. A box has 4 red and 2 white pens. You take two pens without replacement. What is the probability that both are white? Answer with a fraction.

Common questions

How do you turn a probability into a percentage?

Divide the numerator by the denominator and multiply by 100. 3/8 = 0.375 = 37.5%.

If I have rolled five ones in a row, is a one less likely next time?

No. The dice has no memory. Every roll has the probability 1/6 of giving a one.

What is the difference between with and without replacement?

With replacement, the probabilities are the same in the second draw. Without replacement, there is one marble fewer: the denominator goes down by one, and so does the numerator if the marble drawn was a favourable one.

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