 In this video, we're going to look at how to use a tree diagram. A tree diagram shows all the possible outcomes of more than one event by following the possible paths along the branches of a tree. Before you begin, it might be worth going over the basics of probability, including and and all rules for multiple events. Tree diagrams utilize the and and all rules of probability for multiple events. If you want the probability of two events both happening, event A and event B, you multiply the probabilities together. If you want to find the probability of one event or another event happening, but not necessarily both, you would add the values together. There are two types of tree diagrams for two types of situations, dependent and independent. Dependent tree diagrams occur when the second probability is affected by the first. For example, if a coloured counter is chosen and not replaced, the second probability is changed. Independent probabilities are completely separate from each other and so the second choice remains the same no matter the first. These types of tree diagrams are often easier for this reason. Let's start by looking at a tree diagram question. See if you can identify if it's dependent or independent. Jack and Jess are both going to work. The probability Jack will be late is 0.3. The probability that Jess will be late is 0.1. What's the probability that only one of them is on time? Is the question dependent or independent? Pause and see if you can figure out which. So this question is an independent question because the lateness of our first person, Jack, doesn't affect the lateness of our second, Jess. Now let's see how we would answer this probability question. If you would like to have a go first, pause now. So firstly, we need to draw out our tree diagram. We have two events, each with two possible outcomes. So our diagram should look like this. We can then input our probabilities onto the branches. Because the likelihood of Jack being late is 0.3, his probability of being on time is 0.7. This is because all the potential outcomes must add up to 1. Pause and work out Jess's likelihood of being on time. As our probabilities must total 1, we subtract 0.1, her lateness probability, giving us a value of 0.9 for being on time. Putting these values onto our tree diagram should look like this. Now we can look at solving the problem using our and and or rules. The question asks us for the probability that only one is on time. There are two branches which satisfy this requirement. To work out the probability of each of these situations occurring, we need to apply our and rule and multiply them together. Because we have two eventualities that work for our question, we now apply the or rule. So to find the probability of exactly one being on time, we add the two values together. Therefore, the probability of only one being late is 0.34. Now we're going to look at dependent. The application of and and or rules here are the same as before, but the probability values will vary as we move along the branches. Here's our question. A bag contains 11 counters. There are seven red and four blue. One is selected at random and not returned. A second is then selected. What's the probability that they are the same color? If you want to have a go at this question, pause now. If you want to see how to set it up and then have a go, we will show you now. To set up this question, we need a tree diagram and values to the branches. Our first stage of the tree diagram should look like this, with only the first values included. We now need to calculate the remaining values, remembering that they will change as these events are dependent. If the first choice is red, the second choice will have one less red available to choose from. This makes the new red possibility six over 10, one less red to pick and one less counter in the bag. If the first counter choice is blue, there are still seven reds to choose from, but again, only 10 counters remaining. We can now add in the remaining blue values, remembering that they must add up to one. And now we need to look at which of our outcomes work for our question. We need two colors the same, so we want the probabilities for red red and blue blue. We use the and rule to find these values, multiplying them together. We then need our all rule because either red red or blue blue will work for us. Adding these values, we get 54 over 110, which we can simplify to 27 over 55. This is our final answer. So there is our guide to tree diagrams. These can be used for multiple events, not just two. And always remember to check and see if the second event is dependent on the outcome of the first. If you liked the video, give it a thumbs up and don't forget to subscribe, comment below if you have any questions. Why not check out our Fuse School app as well? Until next time.