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As we advance through probability, it becomes very apparent that we need to be quite organized with our problems as they become more complex. In this section we will use tree diagrams to help us calculate probabilities for given situations. Tree diagrams are a visual aid that can help us break down a situation and calculate probabilities. There are two key principles that we must observe for all tree diagrams. First of all, to find the total probability for any given branch on a tree, multiply the individual probabilities along that branch. Secondly, the sum of the probabilities from the ends of each branch must total to 1. We will examine several examples of probabilities using tree diagrams in order to solidify our understanding of this concept.
Example 1
At a restaurant, there are two breakfast platters that are served, one featuring pancakes and one featuring eggs. There are also two choices for drinks, milk or juice. Thirty percent of customers choose the pancake platter while 70 percent choose the egg platter. Forty percent of customers choose milk while 60 percent choose juice. Assume the drink choice is independent of platter choice. Build a probability model for this situation by using a tree diagram.
Solution
Step one is to build the tree diagram as shown below. Be sure to label each branch with what it represents and the associated probability. Step two is to calculate the probabilities at the end of each branch. To do this, we multiply the probabilities along each branch. For example, the top branch's value of 0.28 was found by multiplying 0.7 by 0.4.
Table 2.1, shown below, summarizes the data in the tree diagram. There are two critical ideas to pay attention to here. First of all, the probability at the end of each branch is the product of the probabilities on that branch. Secondly, notice that the sum of the four probabilities at the ends of the branches add up to 1. Table 1.1 summarizes these results from our tree diagram and is called a probability model. Notice that the probabilities in the table sum to 1.
Example 2
The Diamonds and the Dusters baseball teams are playing a best-of-three playoff series. The first team to win two games is the winner of the series. Suppose the Diamonds have a 60% chance to win any game they play against the Dusters. Build a tree diagram and a probability model to determine the probability of each team winning the series.
Solution
Notice that in the tree diagram below, not all branches go three games. There are two sets of branches that only go two games. A third game does not need to be played in all situations.
Table 2.2 below gives the probability model based upon our tree diagram. The probability of the Diamonds winning the series is .36+.144+.144=.648 while the probability of the Dusters winning the series is .096+.096+.16=.352. In our solution, notice that each branch is labeled and includes a probability. Once again, we multiply the values along each branch to get the probability at the end of the branch. Notice again that the total of all the probabilities in the probability model sums to 1.
Example 3
You are dealt two cards from a standard deck of 52 cards. What is the probability that the two cards can be classified as a red card and a face card (in either order)?
Solution
Begin by considering the first card. We are concerned primarily with getting a face card and then a red card or a red card and then a face card. In order to reach our goal, the first card could be a red card, a face card, or both a red and a face card. Any other card would result in us not reaching our goal. As a result our tree diagram will have four initial branches as shown below; red (not face), face (not red), red & face, or other.
Once our first set of branches are complete, we look at the second stage. We will examine the red, not face branch in detail (top branch). There are 20 red cards out of our 52 card deck that are not face cards. Once we have that card, we now want to know the chances of getting a face card (assuming we just drew a red card). There are still 12 face cards in the deck but there are now only 51 cards remaining in the deck. We multiply this branch out to get
.There are two other branches we work in a similar fashion as they are the only other branches that help us achieve our goal. Adding these gives us . There is about a 22% chance of getting a red card and a face card.
The key to success when working with tree diagrams and probability models is to work in a neat and organized fashion. Many errors are often due to sloppy work. In addition, another useful suggestion is to make your tree diagrams large enough so that you have plenty of room to work.
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