Question

elliot wrote a computer program that randomly generates a number from 2 to 10. if he runs the program twice, what is the probability that the number generated both times is a prime number?
\\(\frac{2}{15}\\)
\\(\frac{1}{6}\\)
\\(\frac{16}{81}\\)
\\(\frac{9}{20}\\)

Explanation:

Step1: Find total possible outcomes

The program generates a number from 2 to 10. There are 9 possible numbers (2, 3, 4, 5, 6, 7, 8, 9, 10). When run twice, by the fundamental counting principle, the total number of possible outcomes is $9\times9 = 81$.

Step2: Identify prime - numbers in the range

The prime numbers in the range from 2 to 10 are 2, 3, 5, 7. So there are 4 prime numbers.

Step3: Calculate the probability of getting a prime number in one run

The probability of getting a prime number in one run of the program is $\frac{4}{9}$ since there are 4 prime numbers out of 9 possible numbers.

Step4: Calculate the probability of getting a prime number twice

Since the two runs are independent events, the probability of getting a prime number both times is the product of the probabilities of getting a prime number in each run. So $P=\frac{4}{9}\times\frac{4}{9}=\frac{16}{81}$.

Answer:

$\frac{16}{81}$