Skill Qualification Task 5 -- Conditional Probability
The problems to submit are in red at the bottom.
Here is a site with several more examples:
Here is the Gilmore Dining hall problem we looked at before:
Ate at Gilmore
Did not eat at
People surveyed who got violently ill yesterday
People surveyed who did not get violently ill
And repeated with the symbols for your convenience:
We have looked at simple (marginal) probability and joint (and, or) probability so far. Now let's consider CONDITIONAL probab
Suppose we look only at people who ate at Gilmore, highlighted in yellow above.
In that case, what is the probability that
group got sick?
That would be 25/45 or .556
The expression for that example would be: P(S|G) read: "The probability of S given G". This means to ask, "What is the proba
the person selected ate at Gilmore?"
Here's another example: What is the probability that a person did not eat at Gilmore, given the person got sick? Another way o
might be: "If we only look at people who got sick, what is the probability that one of them did not eat at Gilmore?
The expression would be: P(G'|S) and be calculated like this:
P(G' and S)
P(G'|S) = -----------------P(S)
Notice in the case of conditional probab
for the P(G'|S) = 5/30 = .16667
This approach may help understand wh
as meaning we limit our examination to
Then of the people who got sick, what
By the same logic here are a few more…
What is the probability that someone got sick if we only look at the people who ate at Gilmore?
P(S and G)
Independence. Generically speaking we would say that if the P(A) is equal to the P(A|B) then the events A and B are statistical
In the example above we would check for the P(S) being equal to the P(S|G). Is P(S) = P(S|G)? Is .333 = .5556? No, so the a
If they were independent, we would expect the probability of randomly finding a person who got sick in the whole sample to be e
of randomly finding a person who got sick if we only look at people who ate at Gilmore.
For an example of independence consider two tosses of a coin. The chances of getting heads on the second toss are always .
second tosses for which we got heads on the first toss. We know the results of the first toss are independent of the results of t
We may suspect something is going on with the Gilmore dining hall because the chances of randomly finding some who got sic
we look only at the people who ate at Gilmore (given Gilmore) than if we look at everyone in the sample whether or not they ate
However, it is very important that we realize that not being statistically independent doe not mean causality. There may be a ca
but not being independent does not prove a relationship.
Try to do the work for these examples:
What is the probability that someone did not eat at Gilmore, given they did not get sick?
P(G'|S') = =
For the people who did not get sick, what is the probability that one of them ate at Gilmore?
Lets go for the DVD and TV problem from the last SQT:
Did not Purchase DVD
Did not purchase
Determine the following probabilities:
What is the probability that a customer purchased a DVD, given that customer did not purchase a TV?
What is the probability that a customer, who we know did not purchase a DVD, purchased a TV?
… and with new numbers to keep it lively…
A survey was taken of 130 college professors in New Hampshire. Forty of the professors taught at New England College. The
Now it was also learned that forty of these professors in New Hampshire were homicidal maniacs.
Of the forty professors who taught at New England College thirty were also homicidal maniacs.
Create the table…
Solve for the following probabilities:
What is the probability that a professor is not a homicidal maniac, given that professor does not teach at NEC?
… and the one you are all interested in knowing…
What is the probability that a professor who teaches at NEC is a homicidal maniac?
consider CONDITIONAL probability.
case, what is the probability that someone selected at random from just
means to ask, "What is the probability of someone getting sick, given the
e person got sick? Another way of phrasing the question
ot eat at Gilmore?
in the case of conditional probability it is possible to divide the actual data and get the same answer.
P(G'|S) = 5/30 = .16667
pproach may help understand what's going on with conditional probability. We can think of the word "given"
aning we limit our examination to the given event. In this case we are only looking at people who got sick.
of the people who got sick, what percentage of them did not eat at Gilmore?
he events A and B are statistically independent.
)? Is .333 = .5556? No, so the are not independent.
t sick in the whole sample to be equal to the probability
on the second toss are always .50 even if we only look
ndomly finding some who got sick are greater if
an causality. There may be a causal relationship
stomer did not purchase a TV?
a DVD, purchased a TV?
ht at New England College. The rest taught elsewhere.
Purchase answer to see full