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.
d) How many patients would you expect to vaccinate before the first side effect is observed?

Answer 1

X is what youre interested in

Answer 2

Is x the observed side effects?

Answer 3

no...try to guess a number which were intersted in>?

Answer 4

6% is .06 which is the chance of the side effects. But I dont know what else to do with it

Answer 5

X is the no of patints until the first side effect is observed

Answer 6

okay. I know why it is geometric. so how do i calculate the probabilty of the 1st patient w/ a side effect is the 5th patient?

Answer 7

I'm a little late to the game, but let me think about this one.

Answer 8

So the probability of side effect is 0.06, but probability of nothing is 0.94

Answer 9

So for the nth person to be the one to get the side effect, what must happen is for everyone before them to not get it: \[
(0.94)^{n-1}
\]And then for them to get it: \[
(0.06)
\]Putting this together: \[
(0.06)(0.94)^{n-1}
\]
A geometric series is one of the form: \[
a_n = a_1r^{n-1}
\]This is our verification.

Answer 10

Our probability equation is: \[
\Pr(n) = (0.06)(0.94)^{n-1}
\]

Answer 11

Our expected value is going to be: \[
\sum_{n=1}^\infty n\Pr(n)
\]