How to Use Probability Trees

By Mary Jane Sterling

You will definitely encounter probability or chance problems in a finite math course. One way you can solve these problems is to organize the data visually in probability trees.

For example, say that you’re doing a market survey concerning pizza establishments and the preferences of consumers. You went to Dominick’s Den, Pizza House, and Mama Joe’s interviewing an equal number of people at each establishment. You’ve determined that someone who has eaten at Dominick’s Den is 60% likely to return there and 30% likely next time to go to the Pizza House instead. (That leaves 10% likelihood of going to Mama Joe’s.) Someone who has eaten at the Pizza House is 70% likely to return there and 15% likely to go to Dominick’s Den the next time. And a customer at Mama Joe’s is 40% likely to return and would be 50% likely to go to Pizza House.

This is a lot of information that is tricky to sort through in this format. You can put all this information in a tree to help with your study and conclusions. In the figure, you see the pizza establishment that was visited, with one-third of the interviewees in each, followed by the percent chance of a visit to that same place or another. Note that the percentages in each section all add up to 100%. No other eateries are in the survey.

What’s the chance someone will go to Dominick’s Den, Pizza House, or Mama Joe’s?

With the probability tree in place, you can make all sorts of conclusions—all based on the dependability of your research, of course. Some questions that might be posed are

  • What is the probability that, if you choose a person surveyed at random, that person has gone to Dominick’s Den three times in a row? First, look at the DD line and follow the probability segment that takes you to DD again, and then again. Multiply the three percentages together:
    Answer: 12%.
  • What is the probability that a person will visit all three establishments? For this question, you have to track six different visit patterns: DD-PH-MJ and DD-MJ-PH in the top grouping, PH-DD-MJ and PH-MJ-DD in the middle grouping, and MJ-DD-PH and MJ-PH-DD in the bottom grouping. Multiply the two percentages together in each of those lines and add them together; and they each get multiplied by
    for the starter:
  • What is the probability that a person never went to Pizza House? To compute this answer, you look at the top section and track the DD-DD-DD, DD-DD-MJ, DD-MJ-DD, and DD-MJ-MJ choices. Then go to the bottom section and track the MJ-DD-DD, MJ-DD-MJ, MJ-MJ-DD and MJ-MJ-MJ choices. Multiplying the decimals corresponding to the percentages, you have