Question

A number is chosen at random from the imtegers 1 to 10, find the probabi that the number is either a prime or a multiple of 2

Answer 1

To solve the problem, we first need to identify the prime numbers and the multiples of 2 within the integers from 1 to 10.

### Step 1: Identify Prime Numbers from 1 to 10
The prime numbers between 1 and 10 are:
- 2
- 3
- 5
- 7

So, the prime numbers are: **2, 3, 5, 7** (a total of 4 prime numbers).

### Step 2: Identify Multiples of 2 from 1 to 10
The multiples of 2 between 1 and 10 are:
- 2
- 4
- 6
- 8
- 10

So, the multiples of 2 are: **2, 4, 6, 8, 10** (a total of 5 multiples of 2).

### Step 3: Count Unique Outcomes
Now we need to find the total number of unique integers that are either prime or multiples of 2.

We can list them:
- Prime numbers: **2, 3, 5, 7**
- Multiples of 2: **2, 4, 6, 8, 10**

Now we combine these sets, ensuring that we do not double count the number 2:
- Combined set: **2, 3, 4, 5, 6, 7, 8, 10**

### Step 4: Count the unique numbers in the combined set
The combined set contains the following unique numbers:
- **2, 3, 4, 5, 6, 7, 8, 10** (a total of 8 unique numbers).

### Step 5: Calculate the Probability
The total number of integers from 1 to 10 is 10. The probability of randomly selecting a number that is either a prime or a multiple of 2 is given by the number of favorable outcomes divided by the total outcomes:

\[
P(\text{prime or multiple of 2}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{8}{10} = \frac{4}{5}
\]

Thus, the probability that the number chosen is either a prime or a multiple of 2 is \(\frac{4}{5}\).