Question:
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .
My approach:
Let n be the set for numbers divisible by 2.
$n={2,4,6….100}$
let o be the set for numbers divisible by 3.
$o={3,6,9..99}$
let set p be the set of numbers divisible by 5.
$p={5,10,15…100}$
Let E be the event that the number is not divisible by 2,3 OR(this word is important) 5.
The objective of the question is to find the probability of this event E.
Let X' denote the complement of any set X.
Let P(X) denote the probability of any event(set) X
So according to me $E=n'\cup o'\cup p'$
$P(E)$ turns out to 0.97
My answer is wrong though, according to the given solution:
$E=n'\cap o'\cap p'$
So I think the wording of the question was wrong,
the question should be:
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 AND 5 is ______.
Instead of:
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 OR 5 is ______ .
Best Answer
$\lnot(n \lor o\lor p) \equiv (\lnot n \land \lnot o \land \lnot p)$ by DeMorgan's.
You misinterpreted the question:
Not divisible by $(x \lor y\lor z)$, means not divisible by x, and not divisible by y, and not divisible by z.
So $E= (n\cup o\cup p)' = n'\cap o'\cap p')$ via DeMorgan's in set theory.