# Intersection of Three events (probability)

• jchiz24
In summary, the conversation discusses the inclusion-exclusion principle and its application to proving that the probability of the intersection of three events is greater than or equal to the sum of their individual probabilities minus two. This can be visualized through a Venn diagram where each area is labeled based on its contribution to the sum. The conclusion is that the probability of the intersection is at least as large as the sum of the individual probabilities minus two.
jchiz24

## Homework Statement

Show that for 3 events A, B, C, the probability P of the intersection of A, B, and C is greater than or equal to P(A) + P(B) + P(C) - 2.

aka: P(A intersection B intersection C) > or = P(A) + P(B) + P(C) - 2

N/R

## The Attempt at a Solution

Use venn diagram, look at areas of venn diagram in terms of complement

What you need is inclusion-exclusion principle. It's as true for probability as for cardinality.

In terms of your Venn diagram, label each area by how many times it is counted in P(A)+P(B)+P(C). The '-2' then subtracts each area twice. What do you conclude?

## 1. What is the probability of three events happening simultaneously?

The probability of three events happening simultaneously, also known as the intersection of three events, can be calculated by multiplying the individual probabilities of each event. For example, if event A has a probability of 0.5, event B has a probability of 0.6, and event C has a probability of 0.7, the probability of all three events happening together is 0.5 * 0.6 * 0.7 = 0.21 or 21%.

## 2. How is the intersection of three events different from the intersection of two events?

The intersection of two events is the probability of both events happening together, while the intersection of three events is the probability of all three events happening together. In other words, the intersection of three events is the combined probability of all possible outcomes that satisfy all three events, whereas the intersection of two events is the combined probability of all possible outcomes that satisfy both events.

## 3. Can the intersection of three events ever have a probability of 1?

Yes, the intersection of three events can have a probability of 1 or 100%. This means that the three events are mutually exclusive, and the outcome must satisfy all three events. For example, if event A, event B, and event C cannot occur at the same time, the probability of all three events happening together is 1.

## 4. How does the inclusion-exclusion principle apply to the intersection of three events?

The inclusion-exclusion principle states that the probability of the union of two or more events is equal to the sum of their individual probabilities minus the sum of the probabilities of their intersections. This principle also applies to the intersection of three events, where the probability of all three events happening together is equal to the sum of the probabilities of each event minus the sum of the probabilities of each pair of events intersecting.

## 5. Can the intersection of three events be greater than the probability of each individual event?

Yes, the intersection of three events can have a greater probability than each individual event. This can happen when the three events are not independent of each other, meaning that the outcome of one event affects the outcome of the other events. In this case, the combined probability of all three events happening together can be greater than the probability of each individual event.

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