Probability Calculation for Numbers Divisible by 2 or 5

The question asks for the probability of selecting a number divisible by either 2 or 5 from the set of positive integers up to 200.

Set of Positive Integers Up To 200

The set consists of all positive integers from 1 to 200. The total number of possible outcomes when selecting one integer at random is 200.

Total outcomes = 200.

Numbers Divisible by 2

First, let's find the count of numbers in the set {1, 2, ..., 200} that are divisible by 2. These are the multiples of 2.

The number of multiples of 2 up to 200 is calculated as:

$ \text{Count}( \text{Divisible by 2} ) = \left\lfloor \frac{200}{2} \right\rfloor $ $ \text{Count}( \text{Divisible by 2} ) = 100 $

So, there are 100 numbers divisible by 2.

Numbers Divisible by 5

Next, let's find the count of numbers in the set that are divisible by 5. These are the multiples of 5.

The number of multiples of 5 up to 200 is calculated as:

$ \text{Count}( \text{Divisible by 5} ) = \left\lfloor \frac{200}{5} \right\rfloor $ $ \text{Count}( \text{Divisible by 5} ) = 40 $

So, there are 40 numbers divisible by 5.

Numbers Divisible by Both 2 and 5 (i.e., by 10)

If we simply add the counts for numbers divisible by 2 and numbers divisible by 5, we would be counting the numbers divisible by both (i.e., divisible by 10) twice. We need to find and subtract this count.

The number of multiples of 10 up to 200 is calculated as:

$ \text{Count}( \text{Divisible by 10} ) = \left\lfloor \frac{200}{10} \right\rfloor $ $ \text{Count}( \text{Divisible by 10} ) = 20 $

So, there are 20 numbers divisible by both 2 and 5.

Total Numbers Divisible by Either 2 or 5

To find the total count of numbers divisible by either 2 or 5, we use the Principle of Inclusion-Exclusion:

$ \text{Count}( \text{Divisible by 2 or 5} ) = \text{Count}( \text{Divisible by 2} ) + \text{Count}( \text{Divisible by 5} ) - \text{Count}( \text{Divisible by 10} ) $ $ \text{Count}( \text{Divisible by 2 or 5} ) = 100 + 40 - 20 $ $ \text{Count}( \text{Divisible by 2 or 5} ) = 120 $

Thus, there are 120 numbers between 1 and 200 that are divisible by either 2 or 5. These are the favorable outcomes.

Calculating the Final Probability

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

$ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} $ $ \text{Probability} = \frac{120}{200} $

Now, we simplify the fraction:

$ \frac{120}{200} = \frac{12}{20} = \frac{3}{5} $

Therefore, the probability that the selected number is divisible by either 2 or 5 is 3/5.