17 cards numbered 1, 2, 3, ...., 17 are put in a box and mix thoroughly. A person draws a card from the box. Find the probability that the number on the card is a prime number :

Question

Accepted Answer

To determine the probability that the number on the card drawn is a prime number, we first need to identify the prime numbers between 1 and 17. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

The prime numbers between 1 and 17 are: 2, 3, 5, 7, 11, 13, and 17. This gives us a total of 7 prime numbers.

The total number of cards is 17, as they are numbered from 1 to 17.

The probability of drawing a card with a prime number is calculated by dividing the number of favorable outcomes (prime numbers) by the total number of possible outcomes (total cards). Therefore, the probability is:

$$
\text{Probability} = \frac{\text{Number of prime numbers}}{\text{Total number of cards}} = \frac{7}{17}
$$

This matches Option 3, which is $\frac{7}{17}$.

Now, let's explain why the other options are incorrect:

1. **Option 1: $\frac{9}{17}$** - This option suggests that there are 9 prime numbers between 1 and 17, which is incorrect. As calculated, there are only 7 prime numbers.

2. **Option 2: $\frac{17}{9}$** - This option is not a valid probability because it suggests a probability greater than 1, which is impossible. Probabilities must be between 0 and 1.

3. **Option 4: $\frac{17}{7}$** - This option also suggests a probability greater than 1, which is not possible. Additionally, it implies that there are more prime numbers than total numbers, which is incorrect.

Therefore, the correct answer is indeed Option 3: $\frac{7}{17}$, as it accurately represents the probability of drawing a card with a prime number from the set of 17 cards.