Question

What are the 4 laws of probability?

Answer 1

Hint: Probability theory studies the assignment of probabilities to events. Events are statements just like propositions for which it is not easily possible to say whether they are True or False. If the event cannot happen then, its probability is zero and if it is certain to happen, its probability would be one.

Complete step by step Solution:
The algebra of Events will be a Boolean Algebra just like the algebra of propositions. We will have:
$A \wedge B$ is an event that both $A$ and $B$ will occur.
$A \vee B$ is an event that at least one of $A$ and $B$ will occur.

1. ${A^c}$ that the event will not occur.
To each event, we assign a probability which is a real number between $0$ and $1$. Then, it will satisfy the following laws:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
The probability of the event $A$ or $B$ can be found by adding the probabilities of the separate events $A$ and $B$ and subtracting the intersection of the two events $A$ and $B$.

 2. $P(AandB) = P(A) + P(B|A)$
The probability of the event $A$ and $B$ occurring can be found by taking the probability of the event $A$ occurring and multiplying it by the probability of event $B$ happening given that event $A$ already happened.

 3. Whenever the sample size is less than $5\% $ of the total population, one may treat an event as independent even if they are truly dependent, as it would be rare to select the same item twice.

4. $P(B|A) = \dfrac{{P(AandB)}}{{P(A)}}$
The probability of event $B$ occurring, given that event $A$ has occurred $ = $ the probability of event $A$ and $B$ occurring divided by the probability of event $A$ occurring.

Note:If events $A$ and $B$ are independent, simply multiply $P(A)$ by $P(B)$ . Two events are said to be complementary if they cannot occur at the same time and they make up the whole sample space. If two events are complementary, then their probabilities add up to $1$. If an event is certain to happen, then its probability is one.