Identify Sample Space

The total number of tickets is $50$, so the size of the sample space is $n(S) = 50$.

$n(S) = 50$

Find Multiples of 5

Let $A$ be the set of multiples of $5$ from $1$ to $50$. These are $5, 10, 15, \dots, 50$, giving a total count of $n(A) = \frac{50}{5} = 10$.

$n(A) = 10$

Find Multiples of 10

Let $B$ be the set of multiples of $10$ from $1$ to $50$. These are $10, 20, 30, 40, 50$, giving a count of $n(B) = \frac{50}{10} = 5$.

$n(B) = 5$

Find Intersection (Multiples of both 5 and 10)

The numbers that are multiples of both $5$ and $10$ are the multiples of $10$, which are $5, 10, 15, \dots$ wait, the least common multiple of $5$ and $10$ is $10$. Thus, $A \cap B$ consists of multiples of $10$, so $n(A \cap B) = 5$.

$n(A \cap B) = 5$

Apply Union Formula and Calculate Probability

Using the inclusion-exclusion principle, $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 10 + 5 - 5 = 10$. The probability is $P = \frac{10}{50} = \frac{1}{5}$.

$P = \frac{10}{50} = \frac{1}{5}$