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 o

Answer 1

observed?

Answer 2

@saifoo.khan @Agent_Sniffles @Hero @phi @JamesJ anyone?

Answer 3

So, which parts are you stuck on?

Answer 4

the entire question lol

Answer 5

I got (a). X = number of patients until the first side effect is observed

Answer 6

Yes
So if there is just one patient, what is the value of X?

Answer 7

Ummm.... 1?

Answer 8

btw, I assume this should read:
"Construct a probability distribution table for X up through n [not X] = 5."

Answer 9

how do I verify (b)

Answer 10

So if there is one patient, then you expect one patient should suffer the side effect? No. What is the probability that that one patient suffers the side effect?

Answer 11

Ummmmmm

Answer 12

im not sure how to find that

Answer 13

"There is a 6% chance that a vaccine will cause a certain side effect."

Hence, if n = 1, X = ... ?

Answer 14

6%

Answer 15

?

Answer 16

X(n=1) = 0.06, yes

Now, if there are two patients, what is the probability that the second patient is the first one to experience the side effects?

Pr(2nd patient has side effect) = Pr(2nd patient has side effect | 1st doesn't) Pr(1st does)

Now, what are those two terms on the right side equal to?

Answer 17

**Correction:
Pr(2nd has side effect) = Pr(2nd has side effect | 1st doesn't) . Pr(1st doesn't)

Answer 18

uhh

Answer 19

Pr(2nd has side effect) = probability that 2nd patient is the first to suffer the side effect. Now, what is that equal to?

Answer 20

If the 2nd patient is the first to suffer, then the 1st didn't. Hence
Pr(2nd does) = Pr(2nd does given that the 1st doesn't) TIMES Pr(1st doesn't)

Answer 21

i.e.,
Pr(2nd has side effect) = Pr(2nd has side effect | 1st doesn't) . Pr(1st doesn't)

Make sense?

Answer 22

Where did you go? Can't help you if don't engage in conversation.

Answer 23

@JamesJ im back, sorry

Answer 24

Right.
Does this formula now make sense?
Pr(2nd has side effect) = Pr(2nd has side effect | 1st doesn't) . Pr(1st doesn't)

Answer 25

or being more explicit,
Probability(2nd being first patient to have side effect)
= Probability(2nd has side effect | 1st doesn't) . Probability(1st doesn't)

Answer 26

Ok

Answer 27

So what are the values of
Probability(2nd has side effect | 1st doesn't)
and
Probability(1st doesn't suffer side effects)

Answer 28

I have no clue how to find that

Answer 29

if x=1, n=.06

Answer 30

Well,
Probability(1st doesn't suffer side effects) + Probability(1st does suffer side effects) = 1

What is
Probability(1st does suffer side effects)
?

And therefore what is
Probability(1st doesn't suffer side effects) ?

Answer 31

Its P(0.06?

Answer 32

Probability(1st does suffer side effects) = 0.06, yes

So what is
Probability(1st doesn't suffer side effects) = ... ?

Answer 33

.96?

Answer 34

No
Probability(1st doesn't suffer side effects) = 1 - 0.06 = 0.94
Make sense?

Answer 35

oh yeah thats what i meant oops lol

Answer 36

Now, we want to calculate
Probability(2nd being first patient to have side effect)
= Probability(2nd is first to have side effect | 1st doesn't) . Probability(1st isn't first)

Now, think!

What is
Probability(2nd is first to have side effect | 1st doesn't) ?

Answer 37

Probability(2nd is first to have side effect | 1st doesn't) is just the same as
Probability(2nd having side effect)
as the events are what's called independent events.

What is the value of
Probability(2nd having side effect) ?

Answer 38

ugh im so confused

Answer 39

Just stay with me. What is confusing you right now?

Answer 40

Probability(2nd is first to have side effect | 1st doesn't)

Answer 41

can u just explain to me how to find that

Answer 42

This means what is the probability that the 2nd patient is the first to have the side effect given that (that is what "|" means, "|" = "given that") the first patient hasn't suffered the side effects.

Now
Probability(2nd is first to have side effect | 1st doesn't)
is just equal to the
Probability(2nd patient has side effects)
because whether or not the 1st patient suffers side effects has nothing to do with whether or not the second patient suffers side effects. The drug effects each patient independently, just like separate rolls of dice.

Yes?

If so, what is
Probability(2nd patient has side effects) = .... ?

Answer 43

Well, the probability that ANY patient suffers side effects is 6% or 0.06
Hence
Probability(2nd patient has side effects) = 0.06

Answer 44

Therefore
Probability(2nd being first patient to have side effect)
= Probability(2nd has side effect | 1st doesn't) x Probability(1st doesn't)
= 0.06 x 0.94

Compressing notation now write
Probability(2nd being first patient to have side effect)
as
Pr(2 first) = 0.06 x 0.94

Yes?

Answer 45

Ok!

Answer 46

So what is
Pr(3 first) = ... ?
Expand it out using the | notation

Answer 47

.06 x .94 x ....

Answer 48

Write it out as ...
Pr(3 first) = Pr(3 first | not 1 first and not 2 first) . Pr(not 1 first) . Pr(not 2 first)
= what . what . what

Answer 49

confused, sorry

Answer 50

The probability that the 3rd patient is the first to suffer side effects is equal to

(The probability that the 3rd patient suffers sides effects GIVEN that the 1st and 2nd patients haven't suffered side effects)
TIMES
(The probability that the first patient didn't suffer side effects)
TIMES
(The probability that the second patient didn't suffer side effects)

In notation, that is just
Pr(3 first) = Pr(3 first | not 1 first and not 2 first) . Pr(not 1 first) . Pr(not 2 first)

Answer 51

Im gonna do this prob later i gtg bye