Question

Medical professionals use this distribution to model the probability that a certain number of patients will experience side effects as a result of taking new COVID-19 vaccines. For example, suppose it is known that 5% of adults who take a certain COVID-19 vaccine have negative side effects. Find the probability that more than a certain number of patients in a random sample of 100 will experience negative side effects.

i. P(X>5 patients experience side effects)=
ii. P(X>10 patients experience side effects)=
iii. P(X>15 patients experience side effects)=

**(Hint: This is the probability distribution that is used to model the probability that a certain number of "successes" occur during a certain number of trials.)

2. Website hosting companies use this distribution to model the number of expected visitors per hour that websites will receive. For example, suppose a given website receives an average of 20 visitors per hour. Find the probability that the website receives more than a certain number of visitors in a given hour:
i. P(X>25 visitors)=
ii. P(X>30 visitors)=
iii. P(X>35 visitors)=

**(Hint: This is the probability distribution that is used to model the probability that a certain number of events occur during a fixed time interval when the events are known to

Answer 1

The student's questions relate to using the Binomial distribution for vaccine side effects (fixed number of trials, known success rate) and the Poisson distribution for modeling website traffic (events in a fixed time interval, constant rate).

Step-by-step explanation:

The questions provided involve using probability distributions to model real-world scenarios. Particularly, two distributions are mentioned: the Binomial distribution and the Poisson distribution. The Binomial distribution is used when modeling the probability of a certain number of successes out of a fixed number of trials, where each trial has only two outcomes (success or failure) and the probability of success is constant. This distribution can be applied to modeling vaccine side effects in patients. The Poisson distribution, on the other hand, is useful for modeling the number of events in a fixed interval of time or space when events occur independently and at a constant average rate, such as the number of website visitors per hour.

The student's question regarding the probability of vaccine side effects should be addressed using the Binomial distribution, as it involves a fixed number of trials (100 patients) and a known probability of success (5% side effects). Calculations involve finding P(X > 5), P(X > 10), and P(X > 15) for adverse side effects. For the website visitors, the Poisson distribution applies due to the nature of the events being depicted (average of 20 visitors per hour), thus one would calculate P(X > 25), P(X > 30), and P(X > 35).