Jake runs a simple random number generator that outputs only 00, 11, or 22, each number equally likely. If he runs the simple generator three times, and if one of the numbers is a 11, what is the probability that exactly two of the numbers are 11?

To solve this problem, we need to understand the concept of probability and combinations.

1. First, we know that the random number generator can output three possible numbers: 00, 11, and 22. Each number is equally likely, so the probability of getting any one number is 1/3.

2. If Jake runs the generator three times, there are 3^3 = 27 possible outcomes.

3. We are interested in the outcomes where exactly two of the numbers are 11. This can happen in three ways: 11-11-00, 11-11-22, or 11-00-11, 11-22-11, 00-11-11, 22-11-11.

4. The probability of each of these outcomes happening is (1/3) * (1/3) * (1/3) = 1/27 because each run of the generator is an independent event.

5. Since there are six such outcomes, the total probability is 6 * (1/27) = 6/27 = 2/9.

So, the probability that exactly two of the numbers are 11 given that one of the numbers is 11 is 2/9.

To solve this problem, we need to understand the concept of probability and combinations.

Step 1: Identify the total number of outcomes Since the generator can output 3 different numbers (00, 11, 22) and it is run 3 times, the total number of outcomes is 3^3 = 27.

Step 2: Identify the favorable outcomes We are looking for the cases where exactly two of the numbers are 11. This can happen in 3 ways: 11-11-00, 11-11-22, or 11-00-11, 11-22-11, 00-11-11, 22-11-11. So, there are 6 favorable outcomes.

Step 3: Calculate the probability The probability of an event is the number of favorable outcomes divided by the total number of outcomes. So, the probability that exactly two of the numbers are 11 is 6/27 = 2/9.

So, the probability that exactly two of the numbers are 11 is 2/9.