Finding Probabilities of Compound Events
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?
A. [tex]$\frac{2}{15}$[/tex]
B. [tex]$\frac{1}{6}$[/tex]
C. [tex]$\frac{16}{81}$[/tex]
D. [tex]$\frac{9}{20}$[/tex]
Answer (1)
List possible outcomes: 2, 3, 4, 5, 6, 7, 8, 9, 10.
Identify prime numbers: 2, 3, 5, 7.
Calculate probability of prime in one run: 9 4 .
Calculate probability of prime in both runs: 9 4 × 9 4 = 81 16 .
Explanation
Analyze the problem Let's break down this probability problem step by step! First, we need to figure out what numbers our computer program can generate and which of those are prime.
List possible outcomes The program generates numbers from 2 to 10, so the possible outcomes are: 2, 3, 4, 5, 6, 7, 8, 9, 10. That's a total of 9 different numbers.
Identify prime numbers Now, let's identify the prime numbers in that list. Remember, a prime number is a number greater than 1 that has only two divisors: 1 and itself. So, from our list, the prime numbers are 2, 3, 5, and 7. That means there are 4 prime numbers.
Calculate probability of prime in one run Next, we'll calculate the probability of generating a prime number in a single run of the program. The probability is the number of favorable outcomes (prime numbers) divided by the total number of possible outcomes. In this case, that's 9 4 .
Calculate probability of prime in both runs Since Elliot runs the program twice, and each run is independent of the other, we can find the probability of getting a prime number both times by multiplying the probability of getting a prime number in one run by itself. That is, we calculate 9 4 × 9 4 .
Final Calculation 9 4 × 9 4 = 81 16 So, the probability that the number generated both times is a prime number is 81 16 .
Examples
Imagine you're designing a simple game where a player wins if a randomly generated number is prime. Knowing the probability of generating a prime number helps you balance the game's difficulty and rewards. For instance, if the probability of winning (getting a prime) is low, you might offer a bigger reward to keep players engaged. In this case, with numbers from 2 to 10, the chance of getting a prime number twice in a row is 81 16 , which is just under 20%.
