All language subtitles for 007 The Conditional Probability Formula_en

-: Welcome back everyone.

So far we have only mentioned

the term conditional probability.

Without properly explaining what it is.

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.

Lets go back to the coin flipping example.

From the 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 point five

Therefore, P of A equals

P of A, given B.

This is equivalent to saying that

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 the 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 one over fifty two.

However, it increases if we know its a spade.

Since, P of A given B is one over thirteen,

and P of A is over over fifty-two,

we can say that the two events are dependent.

Similarly, because the probability

of drawing our desired card alters

if we know what is a queen.

We can say A and C are also dependent.

Lets formalize these observations with 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 hold true if the probability of

event B is greater than zero only.

And logically so

If P of B is equal to zero,

Then event B would never occur.

Thus, A given B

would not be interpretable.

Now, lets look at the conditional probability formula

more closely

If we compare to the favorable

overall formula we have been using so far

in this course

we can see many similarities.

To satisfy the conditional probability,

we need both event B and A to occur simultaneously.

This suggests that the intersection of A and B

would consist of all favorable 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 fit into place now, doesn't it?

Great.

Once 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

event if the numeric values are equal.

For instance, lets 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, five 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 four of the eighty economics majors,

graduated with distinction.

The second one suggests,

that four out of the eighty students

who graduated with high grades,

completed a degree in economics.

Before we move on to more complicated

interactions between probabilities,

we are going to explore some real life examples

See you in the next video

and thanks for watching.