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Mathematics · Data and probability

Possible outcomes

How many different results can there be when you roll a dice, toss two coins or choose a meal from the canteen? Learn about experiments, outcomes and events, listing outcomes, tree diagrams and the multiplication principle.

  • 15 min
  • 1 simulation
  • Mini quiz

Updated:

Preview

The question

For lunch, the school canteen offers 3 kinds of sandwich (cheese, vegetable, chicken) and 2 kinds of drink (ayran, a salty yoghurt drink, and fruit juice). Kaan wants to choose a different sandwich-and-drink pair every day. For how many days can he eat without repeating a meal? Is there a quick way to find out without writing them all down?

Try it yourself

No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Number line

A dice and a coin together

Drag the point or use the buttons to move in steps of 1. When you reach the right value, press Check.

Prediction

Predict first

How does the number of possible outcomes change when a coin is added to the dice?

It doesn’t matter whether you’re right; the simulation opens once you choose.

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Frequently asked

5 questions

When you toss two coins, why are there 4 outcomes and not 3?

"One heads, one tails" is really two separate outcomes: HT and TH. Imagine painting the coins different colours: red heads with blue tails is different from red tails with blue heads. The correct list is HH, HT, TH, TT, which is 4 outcomes.

With 3 sandwiches and 2 drinks, why do we multiply instead of adding?

Because every sandwich can be paired with every drink. Each sandwich gives 2 menus, so 3 sandwiches give 2 + 2 + 2 = 3 × 2 = 6 menus. Writing 3 + 2 = 5 just counts the sandwiches and drinks separately, not the pairs.

What is the difference between the sample space and an event?

The sample space is all the possible outcomes of an experiment; for a dice roll, E = {1, 2, 3, 4, 5, 6}. An event is the outcomes you are interested in; the event "the dice shows an even number" is A = {2, 4, 6}. Every event is a subset of the sample space.

What is a tree diagram for?

It helps you list the outcomes in order without missing any. Branches for the first choice grow from the trunk, and branches for the second choice grow from each of those. Counting the leaves at the tips gives the total number of outcomes, the same number the multiplication principle gives.

Does the multiplication principle work for more than two choices?

Yes, you multiply the numbers of options for every choice: m × n × r and so on. For example, with 4 T-shirts, 3 pairs of trousers and 2 pairs of shoes you can make 4 × 3 × 2 = 24 different outfits.