Probability concepts sometimes appear to be a bit abstract. However, they are quite practical as well. Knowing them helps a lot to see the most important structure in seemingly complex problems and express your thoughts clearly. Therefore you should learn to recognize and describe randomness in practical situation as experiments with trails, outcomes and events that make up a sample space. Here I will present a number of exercises that you can use to further your understanding of basic probability concepts with help of tree diagrams. Imagine the situation where you pick up three shells at random at the beach, where there are only two types of shell around, which I'll call Q and R. Can you describe this activity with the terms experiment, trial, outcome, random variable, and event? Try to explain to which part of the shell collecting activity each turn applies. The whole enterprise, picking three shells at random, is the experiment. Each time you pick up a shell is a random trial. So the experiment consists of three trials. Each trial leads to an outcome, you will have picked up, either a shell of that Q or R. At the end of the experiment, you will have outcomes that consists of combinations of three shells, for example QQR or RQR. You can define any outcome or combination of outcomes as an event. For example, all cases where you have picked up at least one Q shell. The random variable in this experiment and also in each trial of this experiment is the type of shell you will pick up. Can you list the sample space for this experiment? The sample space of an experiment comprises all the possible outcomes. In this case, the following eight outcomes are possible. Try to draw a tree diagram that shows the structure of this experiment, with all its outcomes. Your tree diagram for the entire experiment should resemble this one. You have three levels of nodes. The first, the second, and the third time you pick up a shell. At each node, you have one branch which represents picking up a Q shell and one for an R shell. At the end of the branches the final outcomes are listed. Now that you've visualized the experiment in the tree diagram can you define two disjoint events for the outcome of this experiment that are not each other's complement? An example of this would be event A that you would only have picked up one R shell in total, in event B, that you would have picked up two R shells in total. These are disjoint, but not each other's complement, because there is some part of the sample space that belongs to neither of the two events. Can you also give an example of an event for the outcome of this experiment and its complement as well? An example of complementary events could be event A where you would have picked up two or more R-shells versus B of picking up one or zero R-shells in total. Now consider the event that the first shell you pick up would be an R shell and another event at the last shell to pick up is an R shell. Can you describe the relation between these two events? Clearly, these two events are not disjoined because they overlap. So these two events are set to intersect each other. The intersection of the two events is a subset of both events. Cases where the first shell you pick up is an R shell and also the third shell you pick up is an R shell. Let's now try to assign probabilities to the various events. Let's assume that R shells are two times as abundant as Q shells, while they both occur in large quantities. What would then be the probability of picking up a Q shell? The probability is found by considering the relative frequency of both shells, which is already given in this problem. One out of three shells is a Q shell, and two out of three shells is an R shell. So the probability to pick up a Q shell is 1/3. Now, you know the probabilities for a single event of picking up a shell. It's possible to put the probabilities for every event in a tree diagram and start to calculate the probabilities for combined events. For example, what would be the probability to pick up at least two Q shells? The event of picking up at least two Q shells contains the sequences QQR, QRQ, RQQ, and QQQ as outcome. The probability for the first three of these sequences is one third times one third times two thirds. Which is two 27th. The probably for the sequence QQQ is one 27th. Adding up these four probabilities gives seven 27th. The case where shells would not be abundant would change the nature of the system considerably. Let's assume that there are only for Q shells and six R shells at the entire beach, but that you would still collect three out of these randomly. Now put the probabilities for this experiment in a tree diagram again. This is how your tree diagram should look. When you pick up the first shell, there's a probability of 4/10 to choose a Q shell and a probability of 6/10 to choose an R shell. If your first shell was a Q shell, the subsequent probabilities are three ninth and six ninth to select respectively a Q or R shell. And thereafter the probabilities change again. While the denominators in probabilities decrease from ten to nine and eight shells at each time a shell is picked up, the numerators change depending on how many shells of that type are still left on the beach. Can you, for this new situation with only a handful of shells, also calculate the probability to pick up at least two Q shells? While the probabilities in the tree diagram have changed, its structure is still the same. The event of picking up at least two Q-shells still contains the sequences QQR, QRQ, RQQ and QQQ as outcomes. The probability for each of these is given here and the total probability to get any of these outcomes is one third. This was the last exercise I had in stock for you. I hope that it showed you how to effectively apply three diagrams for calculating probabilities and help you to get on grips with the various probability concepts.