Learning Resources
Lesson
You have dealt with Venn Diagrams in earlier grades. We will review their construction and see how they can be used to help calculate the probability of certain situations. The main point to remember is that we are looking at Venn Diagrams primarily as a tool for counting.
Example
In a survey of home technology, out of 100 households asked, 70 said they had a computer, 35 had a DVD player, and 10 had both in their home. Draw a Venn Diagram to represent this information.
Solution
In the above example, suppose we let "own a computer" be event A and "own a DVD player" be event B. Let's use the Venn Diagram to help calculate some probabilities associated with these two events.
From the given information , the last probability, P(A or B), is not obvious . We need to refer to the Venn Diagram to get a count of the number that satisfies the condition A or B. The temptation is to try to calculate P(A or B) by adding P(A) to P(B). Doing that for this example we get:
Clearly this is different than P(A or B) calculated above and in fact is impossible since any probability cannot be greater than 1. Thus we can say for the above example that:
But could there be events A and B such that P(A or B) = P(A) + P(B)?
Example
In a survey of 100 people, it was found that 10 were over-weight and 2 were under-weight. Draw a Venn Diagram to represent this information.
Solution
Since a person cannot belong to both groups, it is clear that the two areas of the Venn Diagram representing them do not overlap.
In the above example, suppose we let "is over-weight" be event A and "is under-weight" be event B. Let's use the Venn Diagram to help calculate some probabilities associated with these two events.
We refer to the Venn Diagram to get a count of the number that satisfies the condition A or B and the condition A and B. In this situation, if we calculate the P(A or B) by adding P(A) to P(B) we get:
For this situation, this is the same as P(A or B) calculated above.
So when can we add P(A) to P(B) to get P(A or B)?
The answer can be seen by close inspection of the Venn Diagrams. If there is no overlap of the two events (that is if the two events cannot occur together), as in the second example, we can add the probabilities to get P(A or B). Such events are called mutually exclusive events because the occurrence of one excludes the occurrence of the other.
Activity
- Complete Investigation 5 on page312 in your text.
- Complete the Investigation Questions 21 - 24 on page 312.
- Complete Investigation 6 on page 313.
- Complete the Investigation Questions 25 - 29 on pages 313 & 314.
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
Test Yourself
Out of the 120 Level 3 students in a certain high school, 40 do physics, 30 do chemistry, and 12 do both. What is the probability that if a student is selected at random from the school, they will be doing either physics or chemistry? What is the probability they will be doing neither physics nor chemistry. Are the events "doing physics" and "doing chemistry" mutually exclusive?