Exercise 1 Page 713

We are asked to use diagrams to understand the rules we wrote to find P(AorBorC) when A, B, and C are disjoint or overlapping events. To do so, we will used Venn diagrams. Let's remember the diagrams that we created!

Review of Venn Diagrams

Venn diagrams are often used to represent sets and to illustrate the probabilities of events occurring. Let's look at an example for two events A and B.

In this diagram, each of the circles represents the probability of an event. The overlapped area represents the probability P(AandB). Venn diagrams can also help us understand how to find certain probabilities. Let's consider the P(AorB).

P(AorB) = P(A) + P(B) - P(AandB) When we draw a Venn diagram to illustrate this probability, every part of the diagram should be shaded, but we need to be certain that each part is only included once.

We can see that this probability contains the probabilities P(A) and P(B). We know that P(AandB) is contained in both P(A) and P(B). Because of that, if we add the probabilities P(A) and P(B), we add P(AandB) twice. Because of this, we need to subtract this probability to find P(AorB).

Disjoint Events

Let's look at the Venn diagram of disjoint events A, B, and C.

We can see that these events do not have any overlapping region in the Venn diagram. Therefore, we can find the probability of Aor BorC by adding each individual probability. P(Aor BorC) = P(A) + P(B) + P(C)

Overlapping Events

Let's look at the Venn diagram of three overlapping events.

Since there are more overlapping regions on this diagram, we need to subtract and add some of these to write a rule for P(Aor BorC). Let's write the rule using the diagram.

In this diagram, the number in each region indicates how many times each individual region has been added. We can see that, in the end, each region is added once. Let's write the rule! P(A or BorC) = P(A) + P(B) + P(C) -P(A and B)-P(B and C) -P(A and C) + P(A and BandC)

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Review of Venn Diagrams

Disjoint Events

Overlapping Events