# Video: Representing Sample Spaces with Tree Diagrams

Madison has drawn a tree diagram to represent the sample space of tossing two coins. By extending the tree diagram, or otherwise, find the number of outcomes in the sample space of the experiment of tossing two coins and spinning two spinners: one with 10 equal sections labeled 1 to 10 and one with 12 equal sections labeled 1 to 12.

02:27

### Video Transcript

Madison has drawn a tree diagram to represent the sample space of tossing two coins. By extending the tree diagram or otherwise, find the number of outcomes in the sample space of the experiment of tossing two coins and spinning two spinners: one with 10 equal sections labelled 1 to 10 and one with 12 equal sections labelled 1 to 12.

So we can see Madison’s tree diagram, and we’ve just included the outcomes of tossing the two of coins. We now need to include the outcomes of spinning the two spinners so that we can count the total number of outcomes in the sample space. However, the number of outcomes when spinning each of the two spinners is very large. We’d need 10 branches for the first spinner and then a further 12 branches for the next spinner.

This will be very difficult to represent on our tree diagram. We’ve been told in the question that we don’t have to extend the tree diagram. We can answer this question another way. So let’s think about a different approach. We’re going to use the fundamental principle of counting. What the fundamental principle of counting tells us is if there are 𝑚 outcomes for the first event and then 𝑛 outcomes for the second, then the total number of outcomes for the two events together is the product 𝑚 times 𝑛.

We can already see this illustrated in the tree diagram. There were two outcomes for the first coin, heads or tails, and two outcomes for the second, also heads or tails. There are four possible outcomes in the sample space of tossing both coins: head head, head tail, tail head, and tail tail. This illustrates the fundamental principle of counting: two multiplied by two is equal to four.

We can extend this to include as many events as we want. So in this case, we have tossing the two coins and then spinning the two spinners. There are two possible outcomes for each of the two coins. There are 10 possible outcomes for the first spinner and 12 possible outcomes for the second. So overall, the number of outcomes will be two multiplied by two multiplied by 10 multiplied by 12.

This is equal to 480. If you prefer to extend the tree diagram, then you could of course do this if you have a big enough piece of paper and enough patience. But using the fundamental principle of counting cuts down the work significantly.