6.3. Probability Using Tree Diagrams and Combinations
6.3. Probability Using Tree Diagrams and Combinations
Probability Using Tree Diagrams and Combinations
In this section, we will apply previously learnt counting techniques in calculating probabilities, and use tree diagrams to help us gain a better understanding of what is involved.
We begin with an example.
Suppose a jar contains 3 red and 4 white marbles. If two marbles are drawn with replacement, what is the probability that both marbles are red?
Solution
Let E be the event that the first marble drawn is red, and let F be the event that the second marble drawn is red.
We need to find
.
By the statement, “two marbles are drawn with replacement,” we mean that the first marble is replaced before the second marble is drawn.
There are 7 choices for the first draw. And since the first marble is replaced before the second is drawn, there are, again, seven choices for the second draw. Using the multiplication axiom, we conclude that the sample space S consists of 49 ordered pairs. Of the 49 ordered pairs, there are 3 × 3 = 9 ordered pairs that show red on the first draw and, also, red on the second draw. Therefore:
Further note that in this particular case:
If in the previous example, the two marbles are drawn without replacement, then what is the probability that both marbles are red?
Solution
By the statement, “two marbles are drawn without replacement,” we mean that the first marble is not replaced before the second marble is drawn.
Again, we need to find
.
There are, again, 7 choices for the first draw. And since the first marble is not replaced before the second is drawn, there are only six choices for the second draw. Using the multiplication axiom, we conclude that the sample space S consists of 42 ordered pairs. Of the 42 ordered pairs, there are 3 × 2 = 6 ordered pairs that show red on the first draw and red on the second draw. Therefore,
Here 3/7 represents
P(
E), and 2/6 represents the probability of drawing a red on the second draw, given that the first draw resulted in a red. We write the latter as
P(Red on the second | red on first) or
. The “|” represents the word “given.” Therefore:
The above result is an important one and will appear again in later sections.
We now demonstrate the above results with a tree diagram.
Suppose a jar contains 3 red and 4 white marbles. If two marbles are drawn without replacement, find the following probabilities using a tree diagram.
a. The probability that both marbles are white.
b. The probability that the first marble is red and the second white.
c. The probability that one marble is red and the other white.
Solution
Let R be the event that the marble drawn is red, and let W be the event that the marble drawn is white. We draw the following tree diagram:
Although the tree diagrams give us better insight into a problem, they are not practical for problems where more than two or three things are chosen. In such cases, we use the concept of combinations that we learned in Chapter 5. This method is best su