Question

Given P(A) = 0.6 and P(B) = 0.3

If A and B are mutually exclusive events, compute P(A or B).

If P(A and B) = 0.2, compute P(A or B).

If A and B are independent events, compute P(A and B).

If P(B|A) = .1, compute P(A and B).

Answer 1

Concepts and reason

Probability:

The ratio of the number of favorable outcomes to certain event and total number of possible outcomes is called as the probability of an event.

Union of two events:

The set of the outcomes that belong to either the two events or any of the two events is called as union of two events.

Intersection of two events:

The set of the outcomes that belong to both the two events s is called as intersection of two events.

Independent events:

Two events A and B are said to be independent events if the occurrence of one event does not affect the occurrence of another event. In other words, if occurring of event A does not influence the event B occurring, then the two events A and B are independent.

Mutually exclusive events:

Let A and B be two events. If two or more events are not occurring at the same time then the events are called as mutually exclusive. Any two events,A and B are said to be mutually exclusive events, when $\left( {A \cap B} \right) = 0$ .

Conditional probability:

The probability of happening of an event given that another event has already happened is called as the conditional probability.

Fundamentals

The probability of an event A is defined as,

Define A and B as two events, the addition rule for the two events is defined as,

$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$

Let A and B be the two events, then

• Multiplication rule for independence of two events: $P\left( {A \cap B} \right) = P\left( A \right)P\left( B \right)$

• The additive rule for mutually exclusive events: $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)$

Conditional probability:

Let A and B be two events, then the conditional probability for event B, given that event A has already happened, can be defined as,

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

The value of $P\left( {A{\rm{ or }}B} \right)$ when the events A and B are mutually exclusive is obtained below:

From the information, $P\left( A \right) = 0.6,P\left( B \right) = 0.3$ .

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right) = 0.6 + 0.3\\\\ = 0.9\\\end{array}$

The value of $P\left( {A{\rm{ or }}B} \right)$ when $P\left( {A{\rm{ and }}B} \right) = 0.2$ is obtained below:

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right) = 0.6 + 0.3 - 0.2\\\\ = 0.9 - 0.2\\\\ = 0.7\\\end{array}$

The value of $P\left( {A{\rm{ and }}B} \right)$ when the events A and B are independent is obtained below:

The required probability is,

$\begin{array}{c}\\P\left( {A \cap B} \right) = 0.6 \times 0.3\\\\ = 0.18\\\end{array}$

The value of $P\left( {A{\rm{ and }}B} \right)$ when $P\left( {B|A} \right) = 0.1$ is obtained below:

The required probability is,

$\begin{array}{c}\\0.1 = \frac{{P\left( {A \cap B} \right)}}{{0.6}}\\\\P\left( {A \cap B} \right) = 0.1 \times 0.6\\\\ = 0.06\\\end{array}$

Ans:

The value of $P\left( {A{\rm{ or }}B} \right)$ when the events A and B are mutually exclusive is 0.9.