The probability of meeting at least one person with the flu in 11 random encounters can be calculated using the complement rule of probability.

Step-by-Step Explanation:

1. Infection Rate:

   • 55% of the population has the flu, so the probability of a single random person not having the flu is: $P(\text{no flu}) = 1 - 0.55 = 0.45$

2. 11 Random Encounters:

   • For 11 independent random encounters, the probability that none of the 11 people encountered have the flu is: $P(\text{no flu in all 11}) = (0.45)^{11}$

3. At Least One Person Has the Flu:

   • The probability of meeting at least one person with the flu is the complement of the probability that none of the 11 people encountered have the flu: $P(\text{at least one flu}) = 1 - (0.45)^{11}$

Let's calculate this value.

The probability of meeting at least one person with the flu in 11 random encounters on campus is approximately 99.98%.

Would you like more details or have any questions about this? Here are some related questions:

1. How would the probability change with fewer or more random encounters?
2. What if the infection rate were lower, say 40%, instead of 55%?
3. How can we calculate the expected number of people with the flu in 11 encounters?
4. What is the probability of meeting exactly one person with the flu in 11 encounters?
5. How can we model this situation using a binomial distribution?

Tip: The complement rule is often a useful tool when calculating "at least one" probabilities.