c Use a tree diagram to find the number of possible outcomes.
D
d Do all second stage events have the same number of possible outcomes?
A
a
B
b
C
c 20
D
d No, see solution.
Practice makes perfect
a We are given a sequence of events and we want to draw a tree diagram to represent a sample space of the described experiment. Let's start with the first stage of the experiment, which is spinning Spinner 1 that has 4 colors on it. First we should draw 4 branches, each representing one color we can spin.
Now we are given that if we spin red, then in the next step we flip a coin. This means that we need to draw two branches from the branch that represents spinning red. One of them will represent flipping heads, and the other one will represent flipping tails.
Next, if we spin yellow then we will roll a die in the next step, and if we spin green we will roll a number cube. Since both a die and a number cube have possible outcomes that are integers from 1 to 6, we should draw 6 branches from the branches representing spinning yellow and green.
Finally, if the result is blue we will spin Spinner 2, which has 6 different colors on it. Therefore, we should draw six more branches that start in a branch representing spinning blue.
b We are asked to draw a Venn diagram to show the possible outcomes of the experiment. Each set will represent possible outcomes after spinning a particular color using Spinner 1. Notice that these sets will not have overlapping regions because they are mutually exclusive.
c To find the number of possible outcomes we can use the tree diagram we made in Part A.
Notice that the number of possible outcomes will be the number of the second (the last) branches.
2+ 6+ 6+ 6 = 20
There are 20 possible outcomes in this experiment.
The number of possible outcomes in a sample space can be found by multiplying the number of possible outcomes from each stage or event.
In other words, when we use this principle we multiply the number of possibilities of every stage. Now, let's notice that in our experiment we have 4 possibilities in the first stage but in the second stage the number of possible outcomes is not the same for every color.
This means that we cannot use this principle to determine the number of outcomes for this experiment. We could do that only if each second stage event had the exact same number of possible outcomes.
