How to Do Probability Tree Diagrams


Probability tree diagrams offer a way of visualizing all the possible outcomes of an event, or a series of events. They especially come in handy if you're having trouble remembering which numbers you must multiply to calculate the probability of a series of events taking place. Probability diagrams are always read from left to right, and can be applied to both independent and dependent events.

Single-Event Diagrams

  • To create a probability tree diagram for one event, first write down a name or symbol to identify the event. For example, you might write "coin toss" or "dice roll." Then draw one line to the right of that symbol for each possible outcome. A coin toss would have two lines because there are two possible outcomes: heads and tails. Put the name of each outcome at the far end of its line.

    Rolling one six-sided die would have six possible outcomes: 1, 2, 3, 4, 5 or 6. For that event, you would draw six lines to the right, angling each so you have enough room to label each "arm" with its associated outcome.

Assigning Probabilities

  • Label each arm of the probability "tree" you just created with the probability of its outcome, expressed either as a decimal or a fraction. If you're doing word problems, you will be given this information or enough clues to figure it out. If you're working with a real-world problem, you'll use either observation, simple logic, or already known information to figure the probabilities out.

    In the case of coin tosses and dice rolls, simple logic is enough. Since the coin has just two sides and is not biased toward either side, each side has a 50 percent chance of coming up. You can express that as a decimal -- 0.5 -- or as a fraction, saying that there is a 1 in 2 chance or 1/2 chance of each face coming up. Label each arm with the appropriate decimal or fraction; you can use either type of expression, as long as you stay consistent through the entire diagram.

    For a dice roll, each face has a 1 in 6 or 1/6 possibility of coming up. In this case the fraction is easier to deal with because if you express it as a decimal, it becomes a repeating decimal (.166...), which is harder to work with.

Multi-Event Diagrams

  • To create probability tree diagrams that address multiple events, just treat each individual outcome as its own event and create another diagram that branches out to the right of it. Again, every outcome should be labeled and every "arm" that represents an outcome should be labeled with the probability of it taking place.

    When you add all possible outcomes for each event, the total should always be 1. If it's not 1, you either missed a possible outcome or did not label their probabilities correctly.

Calculating Probabilities

  • Here's where probability tree diagrams really pay off. Aside from helping you visualize the possible outcomes of any event or any chain of events, they also make it easy to calculate probabilities for an entire sequence of events.

    To do this, take a highlighter and highlight the sequences of outcomes whose probability you want to calculate, starting from the first event on the left and continuing as far as you like to the right. Then multiply together the probabilities for each arm you highlighted. The result is the probability of that entire sequence of events taking place. Once you get used to how this works, you can skip the highlighting and just do the multiplication in your head.

    For instance, if you were calculating probabilities for coin tosses, you might highlight the "heads" outcome for the first event, then highlight the "tails" outcome for the next event beyond that, then "heads" for the third events and so on. Each of those outcomes has a .5 or 1/2 chance of happening. To find the probability of getting heads, tails, heads in sequence, you multiply .5 by .5 by .5, or 1/2 by 1/2 by 1/2, to get a final probability of .125 or 1/8, which you would read as 1 in 8.

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