An integer is chosen at random from the first fifty positive integers. What is the probability that the integer chosen is a prime or multiple of $4$.
An integer can be chosen from first fifty positive integers in $50$ ways.
$\therefore \;$ Number of elements in sample space $S = n\left(S\right) = 50$
Let $A$ be the event of selecting an integer which is a prime number OR a multiple of $4$.
There are $15$ prime numbers between $1$ and $50$
$\left(2, \; 3, \; 5, \; 7, \; 11, \; 13, \; 17, \; 19, \; 23, \; 29, \; 31, \; 37, \; 41, \; 43, \; 47\right)$
$\therefore \;$ $1$ prime number can be selected at random from the first fifty positive integers in ${^{15}}{C}_{1} = 15$ ways
There are $12$ numbers which are multiple of $4$ between $1$ and $50$
$\left(4, \; 8, \; 12, \; 16, \; 20, \; 24, \; 28, \; 32, \; 36, \; 40, \; 44, \; 48\right)$
$\therefore \;$ $1$ number which is a multiple of $4$ can be selected from $12$ numbers in ${^{12}}{C}_{1} = 12$ ways
$\therefore \;$ Number of ways in which an integer can be chosen which is a prime number OR a multiple of $4$ $= 15 + 12 = 27$ ways
$\therefore \;$ Number of elements in $A = n\left(A\right) = 15 + 12 = 27$
$\therefore \;$ Probability of $A = P \left(A\right) = \dfrac{n\left(A\right)}{n\left(S\right)} = \dfrac{27}{50}$