How to Draw a Tree Diagram for Probability

In this explainer, we volition learn how to describe tree diagrams representing 2 or more successive experiments and use them to calculate probabilities.

Retrieve that an experiment is a repeatable process with a known set of possible outcomes. An experiment can comprise merely a unmarried event, such as rolling a die once, simply the term is more often used to describe a combination of two or more events. The repeatable and predictable nature of these processes ways that they are best analyzed using probability.

When calculating probabilities for experiments involving simultaneous events, Venn diagrams can be a helpful fashion to show the possible outcomes. For example, we could utilize a Venn diagram to work out the probability that a randomly called student in a given schoolhouse year studies both mathematics and history.

Sometimes, however, we need to stand for events that happen one after some other, called successive events. The first result in a sequence of successive events may or may not touch the probabilities of the afterward ones, but either way, we demand a simple means to stand for all the possible outcomes. This will then enable us to calculate probabilities. It turns out that the most user-friendly method for doing this is drawing a tree diagram.

Properties: Tree Diagram

A tree diagram is a ways of representing 2 or more successive events.

Each branch of a tree diagram has its result written at its right-hand end and is labeled with the probability of that outcome.

For example, the tree diagram beneath represents the experiment of flipping a off-white coin twice.

We start on the left with the commencement flip of the money. This is represented by a pair of branches coming from a single point that correspond to the outcomes "heads" and "tails." Since the coin is fair, or unbiased, each consequence has a probability of 0.v, which is shown as the label on each branch. Note that these are the only two possible outcomes of flipping a fair money, and then their probabilities sum to 1.

For the 2nd flip of the coin, we motion correct and draw similar pairs of branches starting from each outcome of the first flip. In this way, we can represent all iv possible outcomes of flipping a fair coin twice, namely "heads, heads," "heads, tails," "tails, heads," and "tails, tails." To calculate the probability of any of these four outcomes, we multiply the probabilities along the branches leading to that result. And then, for case, to calculate the probability of "heads, heads," we multiply the probabilities along the branches highlighted beneath to get .

In this detail case, all four possible outcomes have the same probability of 0.25 considering they give rise to four identical calculations. This volition not happen in full general. However, it is significant that . In other words, the probabilities of all the possible outcomes of the experiment sum to one, which is true for all tree diagrams.

Note that, in this simple example, the two successive events are independent, since the outcome of the start flip of the coin has no upshot upon the probabilities of the 2d flip. Oft, nevertheless, tree diagrams are used to correspond successive events where the outcome of the first issue has an impact on the probabilities of the later event (or events); these are called dependent events. With questions on this topic, information technology is e'er important to read them carefully to work out whether the given events are independent or dependent, as we need to make sure that we assign the correct probabilities to the branches of the tree diagram.

Let us start with an example to check our understanding of how to construct a tree diagram.

Example 1: Cartoon a Tree Diagram

Fady chooses a carte from a standard deck of 52 playing cards. He records whether it is one of the 12 face up cards, that is, king, queen, or jack, or but an ordinary number card. So, without putting the card back in the deck, he chooses some other one. Which tree diagram correctly represents these ii successive events?

Respond

Recall that a tree diagram is a means of representing two (or more than) successive events. Each branch has its effect written at its right-hand end and is labeled with the probability of that effect. From the question wording, we must work out what the two successive events are, together with the probabilities of their associated outcomes. We can and so pick the right tree diagram from the four answer options.

Here, the get-go upshot is choosing a card at random from a standard deck of 52 cards, with the two possible outcomes being either a face up card or a number menu. We are told that this carte du jour is not put back in the deck afterward, and so the second upshot is choosing a card at random from the remaining 51 cards, again with the ii possible outcomes of information technology being either a face menu or a number carte.

For the first event, the probability of choosing a face card is given by

Similarly, the probability of choosing a number card is given by

Reviewing the respond options, we can immediately rule out D considering it has these 2 probabilities the wrong way effectually.

Options A, B, and C all accept the correct probabilities for the outcomes in the kickoff upshot, and so now we consider the 2d event. As the first carte du jour is not put dorsum in the deck, this means that the issue of the first consequence affects the probability of the 2nd event, and then these events are dependent. Since there are now only 51 cards to choose from, our denominator must therefore be 51. Consequently, we can also rule out option A considering it shows the aforementioned pairs of probabilities for both events. This tree diagram would exist correct for the equivalent experiment where the outset card was put back in the deck earlier the second 1 was chosen.

We at present only have options B and C to consider. If the first bill of fare was a face card, and then we would be left with 11 face up cards and 40 number cards out of a total of 51 cards. Thus, for the 2d effect, the probability of choosing a face carte would be and the probability of choosing a number bill of fare would exist

On the other mitt, if the commencement card was a number card, so we would be left with 12 face up cards and 39 number cards out of a total of 51 cards. In this case, for the second issue, the probability of choosing a face card would be and the probability of choosing a number menu would be

Comparing these probabilities with those in the remaining answer options, we meet that B shows pairs of probabilities for the second event that contradict the outcomes in the first consequence. For example, the probability () on the first branch of the second event indicates that there are 12 face cards still remaining after one has already been selected in the first event, which is not the case. Meanwhile, C shows exactly the probabilities obtained in a higher place, all in the correct places.

Nosotros conclude that the correct tree diagram to represent these ii successive events is C.

In the next example, we volition practice analyzing a tree diagram in gild to identify a specific outcome and calculate its probability. In full general, we will use the note to refer to the probability of the result .

Example 2: Using a Tree Diagram to Compute Probabilities

The probability that a randomly selected student reviews for a mathematics test is 0.viii. If a student does review, the probability that they laissez passer the exam is 0.9. If they practise not review, the probability that they laissez passer the examination is 0.25.

Using the tree diagram in a higher place, calculate the probability that a student chosen at random does non review and nonetheless passes.

Reply

Call back that a tree diagram is a means of representing two (or more than) successive events. To summate the probability of an private outcome, nosotros multiply the probabilities forth the branches leading to that outcome.

Observe that this tree diagram shows how the probability of a student passing their mathematics exam is afflicted by whether or not they review, meaning that these two events are dependent. Of course this is not actually surprising, every bit we would look there to be a link between reviewing and passing exams.

If a student does non review for their mathematics exam and still passes it, then the issue of the outset effect is that they do non review and the outcome of the second event is that they pass. Therefore, to calculate the required probability, we must multiply the probabilities along the branches shown beneath.

Hence, we have which is our answer.

The probability that the pupil does non review and still passes the mathematics examination is 0.05.

Tree diagrams as well enable us to answer more than circuitous questions that involve calculating the probability of a gear up of outcomes instead of just a single ane. For example, returning to our simple money-flipping experiment, we might be asked to calculate the probability of getting the aforementioned result from both money flips, which corresponds to the outcomes "heads, heads" or "tails, tails." In situations similar these, we multiply forth the branches to get the probabilities of the individual outcomes and so do the actress step of adding those probabilities together. This procedure is illustrated below.

In the post-obit example, we need to read each role of the question carefully to identify the required effect, or prepare of outcomes, before calculating the associated probabilities.

Example three: Using a Tree Diagram to Compute Probabilities

The given probability tree shows two events: and . is the effect of it raining and is the event of a group of friends playing football.

1. Decide the probability of the friends playing football game and it raining.
2. Determine the probability of the grouping of friends playing football irrespective of whether it rains.

Answer

Recollect that a tree diagram is a style of representing two (or more) successive events. To calculate the probability of an private result, we multiply the probabilities along the branches leading to that outcome. To calculate the probability of a set of outcomes, we summate the probabilities of the individual outcomes within the set and so add together them all together.

In the above tree diagram, notice get-go the annotation and , which ways "not " and "non ." This is called set up notation and is ofttimes used in tree diagrams because many events accept only two possible outcomes: either they happen or they practise not. Hither, since is the effect of information technology raining and is the consequence of a group of friends playing football, so means that it does non rain and means that the group of friends practice not play football.

Role 1

For part 1, nosotros must determine the probability of the friends playing football and it raining. This corresponds to the unmarried outcome " and " shown below.

To calculate the required probability, we multiply the probabilities forth the branches to get

Role 2

For part 2, we need to calculate the probability of the grouping of friends playing football irrespective of whether it rains. This is equivalent to calculating the probability of the 2 outcomes beneath where the friends play football, with or without it raining.

To calculate the probability of each individual outcome, nosotros multiply the probabilities along the branches leading to that effect. Then, we add the 2 results together. Therefore, the required probability is

Observe that nosotros could accept shortened this calculation slightly by substituting directly from part ane, which would requite the same answer.

Our side by side case is a discussion trouble from which we must construct a tree diagram and then utilise information technology to compute some probabilities.

Case 4: Drawing a Tree Diagram and Using It to Compute Probabilities

Shady has entered a tennis tournament and wants to practice as often as possible to prepare for it. Nevertheless, whether his coach, Maged, plays tennis with him on whatsoever given 24-hour interval depends on the weather condition. Based on past information, the probability that it will rain on whatsoever given day is 0.4. If it rains, the probability that Maged volition play tennis with Shady is 0.05. If it does non rain, nonetheless, the probability that Maged will play tennis with Shady is 0.9.

1. Draw a tree diagram to represent Shady's lawn tennis grooming possibilities.
2. What is the risk that on whatsoever given mean solar day it rains and Shady and Maged play tennis?
3. What is the probability that Shady and Maged play tennis on any given twenty-four hour period?

Answer

Part i

Remember that a tree diagram is a manner of representing two (or more) successive events. Reading through the question, nosotros run into that this scenario comprises ii successive events. The first event concerns the weather conditions, namely whether it rains or not on any given twenty-four hours. The second event concerns whether or not Shady and Maged play tennis on that 24-hour interval.

For the offset event, nosotros are told that the probability that it rains is 0.four. Therefore, since the probabilities of the outcomes of whatsoever outcome must sum to 1, the probability that it does not rain must exist .

Notation also that the ii events are dependent considering the outcome of the first consequence affects the probabilities of the outcomes of the second event. If it rains, then the probability of Shady and Maged playing is but 0.05. Hence, the probability of them not playing is . If it does non rain, then the probability of Shady and Maged playing is 0.9. Hence, in this instance, the probability of them non playing is . Writing "rain" and "no rain" for the weather outcomes and "yep" and "no" for the lawn tennis-playing outcomes, we can construct the post-obit tree diagram.

Function 2

To work out the chance that on any given day it rains and Shady and Maged play lawn tennis, we must first place the right outcome (or outcomes) from the tree diagram. As shown below, this corresponds to the unmarried upshot of "pelting" followed by "yep."

Retrieve that to calculate the probability of an individual outcome, nosotros multiply the probabilities along the branches leading to that result. Therefore, the required probability is

Role iii

Finally, to work out the probability that Shady and Maged play lawn tennis on any given day, nosotros must again identify the correct consequence (or outcomes). Shady and Maged playing lawn tennis regardless of the weather weather condition corresponds to the two outcomes of "rain" followed by "aye" and "no rain" followed by "yeah" every bit shown beneath.

Think that to calculate the probability of a set of outcomes, we calculate the probabilities of the individual outcomes within the fix and and so add them all together. Hence, in this case we have

Notice that to shorten this calculation, we could have substituted directly from role two, which would give the same issue.

In our final example, we analyze a tree diagram that represents three successive events. Every bit these scenarios give rise to a large number of possible outcomes, we can sometimes find useful shortcuts to reduce the amount of calculation needed when working out probabilities.

Example 5: Drawing a Tree Diagram and Using It to Compute Probabilities

A pocketbook contains 20 gold coins and 11 silver coins. Iii coins are chosen at random from the bag, without replacement. All of the possible outcomes are illustrated in the tree diagram shown below, where G represents a selected gold money and S represents a selected silver coin.

1. Use the tree diagram to find the probability that all 3 of the chosen coins are gold. Requite your answer as a fraction in its simplest class.
2. Discover the probability that at least 1 of the called coins is argent. Give your answer every bit a fraction in its simplest form.

Answer

Recall that a tree diagram is a way of representing two or more successive events. To calculate the probability of an individual outcome, nosotros multiply the probabilities along the branches leading to that consequence. To calculate the probability of a prepare of outcomes, we summate the probabilities of the individual outcomes within the set and so add them all together.

The experiment shown in this tree diagram comprises iii successive events of choosing a gold or a silver coin at random from a bag. In addition, note that the coins are called without replacement, and then the events are dependent, significant that the effect of each event affects the probabilities of afterwards ones.

Part i

To work out the probability that all 3 of the called coins are gold, we first identify the correct outcome from the tree diagram, every bit shown beneath.

To calculate the probability, we multiply the probabilities along the relevant branches and then simplify the resulting fraction:

Part two

To piece of work out the probability that at least 1 of the chosen coins is argent, we over again need to place the correct result (or outcomes). The phrase "at least" is of import here because it tells us that we need the set of outcomes that feature i, ii, or 3 silver coins. However, since seven of the eight possible outcomes have this property, instead of multiplying along the branches and adding the 7 results together for the required probability, we can utilise the post-obit simple trick.

As the probabilities of all the possible outcomes sum to 1, if nosotros calculate the probability of the single outcome that does non have the property we want, we can and then subtract it from 1 to get our answer. This unmarried upshot is the i where three gold coins (and hence no silvery coins) are chosen, and we already know that from part 1. Therefore, the probability that at least 1 of the chosen coins is silver is given by which is a fraction in its simplest form.

Notation that, as expected, nosotros obtain the aforementioned effect if nosotros sum the probabilities of the seven outcomes where at least 1 silvery coin is selected, as shown below.

Let u.s.a. terminate by recapping some key concepts from this explainer.

Central Points

• A tree diagram is a means of representing two or more successive events. Each co-operative has its outcome written at its right-manus end and is labeled with the probability of that upshot.
• When drawing a tree diagram, we must always check if the events to be represented are independent or dependent then that we assign the right probabilities to the branches. For independent events, the outcome of the outset event has no effect on the probabilities of later ones, whereas for dependent events, the outcome of the first event has an touch on the probabilities of after ones.
• To calculate the probability of an individual effect, we multiply the probabilities along the branches leading to that consequence. To summate the probability of a set up of outcomes, we calculate the probabilities of the individual outcomes inside the set and and then add them all together.
• If an experiment comprises two events and each outcome has two possible outcomes, our tree diagram will look like this.

  

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Source: https://www.nagwa.com/en/explainers/520190598757/

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