Conditional probability is the likelihood of an event occurring, assuming a different
one has already happened.
In this lecture, we are going to show you how to compute and interpret such probability.
Let’s go back to the coin flip example from last lecture.
“A” represents getting heads and “B” represents getting heads on the previous flip.
The probability of getting heads now, after getting heads last time, is still 0.5.
Therefore, P of A, equals P of, A given B.
This is equivalent to saying the two events are independent.
Earlier in the course we also mentioned that if any two events are independent, the probability
of their intersection is the product of the individual probabilities.
Now, let us examine the Queen of Spades example from last lecture, where A represented drawing
that exact card, B represented drawing the correct suit and C represented getting a queen.
Normally, the probability of drawing the Queen of Spades is equal to 1 over 52.
However, it increases if we know it’s a spade.
Since “P of A, given B” is 1 over 13 and P of A is 1 over 52, we can say the two events
are dependent.
Similarly, because the probability of drawing our desired card alters, if we know it is
a queen, we can say A and C are also dependent.
Let’s formalize these observations with a formula.
By definition, the conditional probability of an event A, given an event B, equals the
probability of the intersection of A and B, over the probability of event B occurring.
This holds true if the probability of event B is greater than 0 only.
And logically so.
If P of B is equal to 0, then event B would never occur.
Thus, “A, given B” would not be interpretable.
Now, let’s look at the conditional probability formula more closely.
If we compare it to the “favourable over all” formula, we can see many similarities.
To satisfy the conditional probability, we need both events B and A to occur simultaneously.
This suggests that the intersection of A and B would consist of all our favourable outcomes
for this probability.
Secondly, the conditional probability requires that event B occurs, so the sample space would
simply be all outcomes where event B is satisfied.
Everything fits into place now, doesn’t it?
Great!
Онце we know the intuition behind the formula, we can focus on the importance of
conditional probability.
Firstly, try to remember that the order in which we write the elements for the conditional
probability is crucial.
P of A, given B is definitely not the same as P of B, given A, even if the numeric values
are equal.
For instance, let’s explore the characteristics of Hamilton College’s class of 2018.
Five percent of the students who got a degree in Economics, graduated with honors.
At the same time, 5 percent of all students who graduated with honors, completed a concentration
in Economics.
These two statements might have the same conditional probability, but they hold completely different
meanings.
In particular, the first one suggests that only 4 of the 80 Economics majors, graduated
with distinction.
The second one suggests that 4 out of the 80 students who graduated with high grades,
completed a degree in Economics.
Many scientific papers rely on conducting experiments or surveys.
They often provide summarized statistics we use to analyse and interpret how certain factors
affect one another.
An example would illustrate this better.
Imagine you conducted a survey where 100 men and women of all ages were asked if they eat
meat.
The results are summarized in the table on the screen.
We see 15 of the 47 women that participated are vegetarian as are 29 out of the 53 men.
Now, if A represents being vegetarian and B represents being a woman, then P of A, given
B and P of B, given A express different events.
The former equals 15 over 47 and represents the likelihood of a woman being vegetarian,
while the latter equals 15 over 44 and expresses the likelihood of a vegetarian being a woman.
Since 15 over 44 is greater than 15 over 47, it is more likely for a woman not to eat meat,
than for a vegetarian to be female.
That’s quite interesting and shows you that in probability theory things are never straight-forward!
Now that you understand the distinctions between the different conditional probabilities of
two events, we can introduce an important concept – the law of total probability.
Okay!
Imagine A is the union of some finitely many events “B one”, “B two” and so on.
This law dictates that the probability of A is the sum of all the conditional probabilities
of A, given some B, multiplied by the probability of the associated B.
To clear any confusion, let us go back to the survey example.
The probability of being vegetarian equals the sum of the probability of being a vegetarian,
given the person is male, times the probability of being male, and the probability of being
a vegetarian, given the person is female, multiplied by the probability of being female.
If we plug in values from the table, we get the probability of being vegetarian, equals
29 over 53, times 53 over 100 plus 15 over 47, times 47 over 100.
That’s 44 over 100, or 0.44.
Therefore, according to the survey, there is a 44% chance of someone being vegetarian.
We can refer back to the table and confirm the probability of being vegetarian is indeed
0.44, so our answer checks out.