Question

Assume that we have two events, A and B, that are mutually exclusive. Assume further that we know P(A) = 0.30 and P(B) =0.40.

1. What is P(A B)?
   
   
2. What is P(A | B)?
   
   
3. Is P(A | B) equal to P(A)?
   
   
   Are events A and B dependent or independent?
   
   
4. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Is this statement accurate?
   
   
5. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Answer 1

Concepts and reason

The probability is the number that reflects the chance or likelihood that a particular event will occur. The probabilities can be expressed as proportions that range from 0 to 1 and also they can express as percentages from 0% to 100%.

The multiplication rule applies to the situation when we want to know the probability of the intersection of two events. That is, the probability that two events both occur.

The conditional probability is the event given the additional information that some other event has already occurred.

If the outcome of any event does not affects the possible outcomes of the other events then the events are independent.

If the happening of any of them precludes the happening of all the others, then the events are mutually exclusive.

Fundamentals

Consider A is the event.

Consider B is another event.

The events A and B are mutually exclusive.

The formula for the conditional probability of the events A given by event B as follows:

$P\left( {A|B} \right) = \frac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}$

If the two events are mutually exclusive then the intersection of the two events is empty.

(1)

The summary statistics as follows:

$\begin{array}{c}\\P\left( A \right) = 0.30\\\\P\left( B \right) = 0.40\\\end{array}$

Here, the two events A and B are mutually exclusive events.

Therefore,

$P\left( {A \cap B} \right) = 0$

(2)

Therefore,

$\begin{array}{c}\\P\left( {A|B} \right) = \frac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\\\\ = \frac{0}{{0.40}}\\\\ = 0\\\end{array}$

Hence, the required conditional probability is,

(3)

$\begin{array}{c}\\P\left( {A|B} \right) = \frac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\\\\ = \frac{0}{{0.40}}\\\\ = 0\\\end{array}$

Therefore,$P\left( {A|B} \right) \ne P\left( A \right)$

[Part 3]

(4)

No.

The two events are mutually exclusive then they are not independent.

That is, the statement is wrong.

[Part 4]

(4)

From the results of the part (1) and part (2), observe that the mutually exclusive events are not independent events.

That is, the mutually exclusive events are dependent events.

[Part 4]

Ans: Part 1

The probability of the intersection of the two events A and B is 0.

Part 2

The conditional probability of the event A given by the event B is, 0.

Part 3

The two events are neither dependent nor independent.

Part 4

The mutually exclusive events are not independent events.

Part 4

The general conclusion is the mutually exclusive events are dependent events.