Table of Contents

## How do you prove P AUB?

P(AUB) = P(A) + P(B)- P(A∩B). Note: If A and B are any two mutually exclusive events then P(A∩B)=0. Then P(AUB) = P(A)+P(B).

## How do you prove two events are mutually exclusive?

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0….If G and H are independent, then you must show ONE of the following:

- P(G|H) = P(G)
- P(H|G) = P(H)
- P(G AND H) = P(G)P(H)

**How do you prove two events are independent?**

Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.

**How do you prove disjoint events?**

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive. Put in formal terms, events A and B are disjoint if their intersection is zero: P(A∩B) = 0.

### How do you find P AC B?

Alternatively, P(Ac∩B) = P(Ac|B)P(B) = (1−P(A|B))P(B) = (1−0.4)×0.5 = 0.3. (b) By the total probability formula, P(A) = P(A|B)P(B)+P(A|Bc)P(Bc)=0.2×0.8+ 0.3 × (1 − 0.8)= 0.22.

### How do you prove that events A and B are independent?

The events A and B are independent if P (A ∩ B) = P (A) P (B). Proof: From the definition of an independent event, we have P (A | B) = P (A) ⇒ P (A ∩ B) ⁄ P (B) = P (A) or, P (A ∩ B) = P (A) P (B). Here, P (B) ≠ 0.

**How do you prove that events A and B are mutually exclusive?**

If A and B are independent events, then the events A and B’ are also independent. Proof: The events A and B are independent, so, P (A ∩ B) = P (A) P (B). From the Venn diagram, we see that the events A ∩ B and A ∩ B’ are mutually exclusive and together they form the event A. A = ( A ∩ B) ∪ (A ∩ B’). Also, P (A) = P [ (A ∩ B) ∪ (A ∩ B’)].

**How do you find the probability of two independent events?**

Theorem 1 : If A and B are two independent events associated with a random experiment, then P (A⋂B) = P (A) P (B) Probability of simultaneous occurrence of two independent events is equal to the product of their probabilities.

#### What is P(AUB) = P(B)?

P(AUB) >= P(B). Probability is analogous to area or volume or mass. Consider the unit square, which has length unity on each side. Its total area is 1 (= 100\%). Let’s call the square S, just like outcome space.