Final answer:
To find the probability that nobody will be denied boarding, we can use the binomial probability formula. We can also estimate the probability using a normal approximation to the binomial distribution. The exact probability can be calculated using the formula, while the estimated probability can be found using the normal distribution.
Step-by-step explanation:
To find the probability that nobody will be denied boarding, we can use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
In this case, we want to find the probability of getting exactly 0 customers who don't show up. So k = 0. The number of combinations of 80 tickets taken 0 at a time is simply 1. The probability of a customer not showing up is 0.05, so p = 0.05. And the number of trials is 80.
Using the formula:
P(X = 0) = C(80, 0) * 0.05^0 * (1-0.05)^(80-0)
P(X = 0) = 1 * 1 * 0.95^80
Calculating this value yields the exact probability that nobody will be denied boarding. To estimate the probability using the law of small numbers, we can use a normal approximation to the binomial distribution. The mean of the binomial distribution is np and the standard deviation is sqrt(np(1-p)). In this case, n = 80 and p = 0.05.
The mean of the binomial distribution is 80 * 0.05 = 4, and the standard deviation is sqrt(80 * 0.05 * (1-0.05)) = 1.8. Since we are looking for the probability that nobody will be denied boarding (0 customers not showing up), we can use the normal distribution to estimate this probability as:
P(X = 0) = P(Z < (0.5 - 4) / 1.8)
Calculating this value using the standard normal table or a calculator gives us the estimated probability.