The probability of choosing a prime number when a number is chosen at random from 1 to 15 is 6/15, which simplifies to 2/5.
Understanding this probability involves identifying prime numbers within the given range and applying the fundamental principles of probability.
Understanding Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This definition is crucial for correctly identifying which numbers in a set are prime. For example, 2 is prime because its only divisors are 1 and 2. The number 4 is not prime because it can be divided by 1, 2, and 4. The number 1 is a special case and is not considered a prime number.
Within the range of 1 to 15, we can list all numbers and identify which ones fit the definition of a prime number:
Number | Divisors | Prime? |
---|---|---|
1 | 1 | No |
2 | 1, 2 | Yes |
3 | 1, 3 | Yes |
4 | 1, 2, 4 | No |
5 | 1, 5 | Yes |
6 | 1, 2, 3, 6 | No |
7 | 1, 7 | Yes |
8 | 1, 2, 4, 8 | No |
9 | 1, 3, 9 | No |
10 | 1, 2, 5, 10 | No |
11 | 1, 11 | Yes |
12 | 1, 2, 3, 4, 6, 12 | No |
13 | 1, 13 | Yes |
14 | 1, 2, 7, 14 | No |
15 | 1, 3, 5, 15 | No |
Based on this analysis, the prime numbers from 1 to 15 are:
- 2
- 3
- 5
- 7
- 11
- 13
There are 6 prime numbers in this range. For more detailed information on prime numbers, you can refer to resources like Wikipedia's page on Prime Numbers.
Calculating Probability
Probability is a measure of the likelihood that an event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
The formula for probability is:
$$P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$
In this specific scenario:
- Total Number of Possible Outcomes: When choosing a number at random from 1 to 15, there are 15 distinct numbers that could be selected. This forms our sample space.
- Number of Favorable Outcomes: As identified above, there are 6 prime numbers (2, 3, 5, 7, 11, 13) within this range. These are the outcomes that satisfy the condition.
Applying these values to the probability formula:
$$P(\text{Prime Number}) = \frac{6}{15}$$
The Probability Result
The probability of choosing a prime number from 1 to 15 is $\frac{6}{15}$. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
$$\frac{6 \div 3}{15 \div 3} = \frac{2}{5}$$
Therefore, the probability is 2/5. This means that for every five numbers chosen randomly from 1 to 15, you would, on average, expect two of them to be prime. Understanding basic probability concepts is fundamental in various fields, from statistics to everyday decision-making. For further learning on probability, consider resources like Khan Academy's Probability Basics.