Q3: What is the probability that you will get two heads on flips three and four,
conditional
on getting
two heads on flips one and two?
A3:
Probability of two heads on
p
(F3&4) conditional on two heads for p(F1&2).
Using Bayes Theorem, we determine that it is not very useful in this form. Here, we are
looking for the probability of the second item,
p
(F3&4), given the first,
p
(F1&2). The
vertical bar means “given.”
Which still is not very helpful. We did not calculate it this way. We did not calculate
p
(F1&2|F3&4). If we had, we would know the answer is just 1 minus the result. But we
did
calculate each part’s relation to the whole.
So
use the form of Bayes’ theorem that
uses the law of total probability as the denominator
:
Which is
p
(F3&4)’s
proportion
to the total probability. From the table we made, we
already know all of those pieces on the right-hand side:
Example Problem
Interpretation:
The
conditional probability
that you will flip two heads in a row after
previously flipping two heads in a row is 50%. Bear in mind that the
raw probability
of flipping two

heads in a row in
independent
trials is 25% and flipping four heads in a row in independent trials is
6.25%.

Recap of the Method Used in Example Problem:
1.
Define success of an individual trial.
2.
Use the binomial equation to calculate the cumulative probability of each part of the system.
3.
Make a table with each part’s
p
and its percentage contribution to the whole. The percentages
must add to 1.
4.
Calculate the
p
(Total) using the law of total probability.
5.
Use the total probability form of Bayes’ theorem to calculate the
conditional probability
for the
item of interest.
6.
Interpret.
Homework Problems:
Using the example problem as reference, complete the following five problems.
Turn in the work and results for each of the five problems. (Note: 4 includes a,b,c and 5 includes a,b,c,d).
Note that each problem provides a hint which should guide you through the problem along with the
Example Problem as a reference.
Post any questions you have to the General Discussion area.
Problem 1:
You are taking a 10 question multiple choice test. If each question has four choices and you
guess on each question, what is the probability of getting one question correct?
[Hint: This is a binomial
in the form of 10 choose 1 with p=.25.]
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