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Probability Lesson 2

this lesson is under construction

For this lesson, I recommend viewing in: Separate Lesson Viewer. Lessons that do not specify this should be viewed in CauseCade.

Conditional Probability

[ex.] Let's dust off our month example again and define two events - which are sets - with one being the months of the year that have 31 days, and the other being the months of the year that have an R in the name

Ω={Jan,Feb,Mar,Apr,May,June,July,Aug,Sept,Oct,Nov,Dec}

L={Jan,Mar,May,Jul,Aug,Oct,Dec}

R={Jan,Feb,Mar,Apr,Sept,Oct,Nov,Dec}

The probability of L, P(L) is 7/12, and P(R)=8/12. This should be apparent.

We determined last lesson that L∩R={Jan,Mar,Oct,Dec}, with probability 4/12=1/3.

Let´s now look at a case where we know something about L and wonder about the probability of R. As an example we will take that someone says that the party is in a long month. We may then be wondering what the chances are that this month contains an r. Let´s call this P(R|L). Naturally, such an occurence should be contained in L∩R (we know it must be in L, and we are only looking for months with r. We could also just find the events from Omega or L and R manually).

We find that that the Probability is 4/7. Now hold on, why is it not 1/3? This is because we already know that the month is long, and are therefore looking only at 7 months. Of those 7 months, only 4 contain an R. The ones that contain and r and are long is precisely those contained in L∩R. We also see that the probability of a month with r that is long is precisely the proportion that P(L∩R) is of (L).

Let´s look at that again.

  • we found that P(L∩R)≠P(R|L) (in general)
  • P(R|L) is the proportion that P(L∩R) is of P(L)

Below I have included a visual representation of this, which I think may clarify this point further.

<(image)>

This brings us to the following statement about conditional probability:

P(A|B)=P(A∩B)/P(B)

notice that the bottom of the fraction contains P(B), not P(A). From the section above it should be apparent that writing P(A) there should be incorrect

Naturally, we can rewrite this as:

P(A∩B)=P(A|B) * P(B)

Law of Total Probability