To find the probability, we can use the Poisson distribution formula. The formula for the probability of exactly x events happening is given by P(X = x) = (e^(-λ) * λ^x) / x!. To find the probability of at least 10 patients arriving, we can use the complement rule. For the probability of another patient arriving before 11:45 am, we can use the cumulative distribution function (CDF) of the Poisson distribution.
In order to solve this problem, we can use the Poisson distribution formula. The Poisson distribution is a discrete probability distribution that gives the probability of a number of events occurring in a fixed interval of time or space.
Probability of exactly 7 patients arriving: Using the Poisson distribution formula, we can calculate this probability by plugging in the appropriate values. In this case, the mean (λ) is equal to 6 patients per hour and the time period (t) is 30 minutes, which is half an hour. The formula for the probability of exactly x events happening is given by P(X = x) = (e^(-λ) * λ^x) / x!
Probability of at least 10 patients arriving: To find the probability of at least 10 patients arriving, we need to calculate the sum of the probabilities of 10, 11, 12, and so on, up to the maximum number of patients possible. We can use the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. So, the probability of at least 10 patients arriving is 1 minus the sum of the probabilities of 0, 1, 2, ..., 9 patients arriving.
Probability of another patient arriving before 11:45 am: Since the patient arrives at 11:30 am, we need to find the probability that another patient arrives before 11:45 am which is 15 minutes later. We can use the cumulative distribution function (CDF) of the Poisson distribution to calculate this probability. The CDF gives us the probability of getting up to a certain number of events. In this case, we want to find the probability of getting 1 or more events (i.e., another patient arriving) before 11:45 am.