### Instruction: In a community, it is known that 20% of the population have tickets to attend a music festival. If 100 people randomly selected from the community, what is the probability that at least 25 of them have tickets to attend the festival?### Response: To solve this problem, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

where:
- P(X = k) is the probability of having exactly k successes (people with tickets) in n trials (people selected)
- C(n, k) is the number of combinations of choosing k successes from n trials, which can be calculated as C(n, k) = n! / (k! * (n-k)!)
- n is the number of trials (in this case, 100 people selected)
- k is the number of successes (in this case, at least 25 people with tickets)
- p is the probability of success (in this case, 20% or 0.2)
- (1-p) is the probability of failure (in this case, 80% or 0.8)

Since we want to find the probability of at least 25 people having tickets, we need to calculate the sum of probabilities for k = 25 to k = 100:

P(X ≥ 25) = Σ P(X = k) for k = 25 to 100

P(X ≥ 25) = Σ [C(100, k) * (0.2)^k * (0.8)^(100-k)] for k = 25 to 100

Now, we can calculate the probabilities for each k and sum them up:

P(X ≥ 25) ≈ 0.9986

So, the probability that at least 25 of the 100 randomly selected people have tickets to attend the festival is approximately 99.86%.