Question

There is a 6% chance that a vaccine will cause a certain side effect. A number of patients are given the vaccine. We are interested in the number of patients vaccinated until the first side effect is observed.

a) Define the random variable X.
b) Verify that this setting is a geometric setting.
c) Find the probability that the 5th patient is the first to experience a side effect.
Construct a probability distribution table for X up through X=5. Please check with your instructor on how to submit your work
d) How many patients would you expect to vaccinate before the first side effect is observed?

Answer 1

The random variable X represents the number of patients vaccinated until the first side effect is observed. This is a geometric setting with a 6% chance of a side effect for each vaccination. The probability that the 5th patient is the first to experience a side effect is approximately 0.03887. The probability distribution table for X up through X=5 is provided.

Step-by-step explanation:

a) The random variable X is defined as the number of patients vaccinated until the first side effect is observed.

b) This setting is a geometric setting because it meets all the characteristics of a geometric distribution: each vaccination is a Bernoulli trial, there is a fixed probability of success (6%), and each trial is independent of the others.

c) To find the probability that the 5th patient is the first to experience a side effect, we use the formula P(X=k) = (1-p)^(k-1) * p, where p is the probability of success and k is the number of trials. In this case, p = 0.06 and k = 5. Plugging in these values, we get P(X=5) = (1-0.06)^(5-1) * 0.06 = 0.03887 (rounded to five decimal places).

d) To construct a probability distribution table for X up through X=5, we can calculate the probabilities using the formula mentioned in part c. The table would look like this:

XP(X)10.0620.056430.053040.049850.0468