WEBVTT
00:00:00.520 --> 00:00:06.320
Madison has drawn a tree diagram to represent the sample space of tossing two coins.
00:00:07.080 --> 00:00:34.960
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.
00:00:36.240 --> 00:00:44.000
So we can see Madison’s tree diagram, and we’ve just included the outcomes of tossing the two of coins.
00:00:44.560 --> 00:00:53.320
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.
00:00:54.040 --> 00:01:00.680
However, the number of outcomes when spinning each of the two spinners is very large.
00:01:00.680 --> 00:01:08.320
We’d need 10 branches for the first spinner and then a further 12 branches for the next spinner.
00:01:09.440 --> 00:01:13.200
This will be very difficult to represent on our tree diagram.
00:01:13.920 --> 00:01:18.200
We’ve been told in the question that we don’t have to extend the tree diagram.
00:01:18.880 --> 00:01:22.600
We can answer this question another way.
00:01:23.560 --> 00:01:26.000
So let’s think about a different approach.
00:01:26.400 --> 00:01:28.720
We’re going to use the fundamental principle of counting.
00:01:29.520 --> 00:01:46.480
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.
00:01:47.480 --> 00:01:50.880
We can already see this illustrated in the tree diagram.
00:01:51.600 --> 00:02:00.880
There were two outcomes for the first coin, heads or tails, and two outcomes for the second, also heads or tails.
00:02:01.360 --> 00:02:13.960
There are four possible outcomes in the sample space of tossing both coins: head head, head tail, tail head, and tail tail.
00:02:14.800 --> 00:02:20.280
This illustrates the fundamental principle of counting: two multiplied by two is equal to four.
00:02:21.280 --> 00:02:26.120
We can extend this to include as many events as we want.
00:02:26.760 --> 00:02:32.480
So in this case, we have tossing the two coins and then spinning the two spinners.
00:02:33.320 --> 00:02:36.600
There are two possible outcomes for each of the two coins.
00:02:37.000 --> 00:02:45.240
There are 10 possible outcomes for the first spinner and 12 possible outcomes for the second.
00:02:45.920 --> 00:02:52.400
So overall, the number of outcomes will be two multiplied by two multiplied by 10 multiplied by 12.
00:02:53.160 --> 00:02:55.640
This is equal to 480.
00:02:56.600 --> 00:03:07.560
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.
00:03:08.280 --> 00:03:12.920
But using the fundamental principle of counting cuts down the work significantly.