We calculate this as 25% chance of flipping two heads on flips one and two. Same thing for
flips three and four.
Let’s now call the first event
p
(F1&2) and the second one
p
(F3&4).
Next,
build the table
. Notice that all the % contributions have to add up to 1.
Table of contributions to total probability:
P
% contrib.
p
(F1&2)
.25
.50
p
(F3&4)
.25
.50
Using the
law of total probability
:
Total probability for two heads then two heads is (.50 x .25) + (.50 x .25) =
25%.
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.