Question

A and B are events such that \[P(A) = 0.42,P(B) = 0.48\] and \[P\left( {A{\text{ }}and{\text{ }}B} \right) = 0.16\]
Determine \[\left( i \right){\text{ }}P\left( {{\text{not }}A} \right)\], \[\left( {ii} \right){\text{ }}P\left( {{\text{not }}B} \right)\]and \[\left( {iii} \right){\text{ }}P\left( {A{\text{ or }}B} \right)\]

Answer 1

Hint: In order to solve this problem you need to know that \[P\left( {{\text{not }}A} \right){\text{ = }}1 - P\left( A \right)\],\[P\left( {{\text{not }}B} \right){\text{ = }}1 - P\left( B \right)\], and \[P\left( {A{\text{ or }}B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A{\text{ and }}B} \right)\].

Complete step-by-step answer:
Now according to question it is given
\[P\left( A \right) = 0.42,P\left( B \right) = 0.48{\text{,}}P\left( {A{\text{ and }}B} \right) = 0.16\]
Then,
Probability of not $A$will be given as,
\[\left( {\text{i}} \right)P\left( {{\text{not }}A} \right){\text{ = }}1 - P\left( A \right) = 1 - 0.42 = 0.58\]
Similarly,
Probability of not $B$will be given as
\[\left( {{\text{ii}}} \right)P\left( {{\text{not }}B} \right){\text{ = }}1 - P\left( B \right) = 1 - 0.48 = 0.52\] and
\[\left( {{\text{iii}}} \right)\]Probability of $A$or $B$will be given as
We know that \[P\left( {A{\text{ or }}B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A{\text{ and }}B} \right)\]
So, \[P(A{\text{ or }}B) = 0.42 + 0.48 - 0.16 = 0.74\]

Note: Probability of an event can neither be negative nor it can be greater than 1. In this type of question first note down all the given details. Afterwards apply the formula of Probability of not $A$. \[P\left( {{\text{not }}A} \right){\text{ = }}1 - P\left( A \right)\], Probability of not $B$\[P\left( {{\text{not }}B} \right){\text{ = }}1 - P\left( B \right)\] and Probability of A or B \[P\left( {A{\text{ or }}B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A{\text{ and }}B} \right)\]. Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.