Conditional probability is the probability of an event occurring, given that another event has occurred. This is written as the probability of B
given that A has already occurred. For instance, lets say that you are drawing 2 cards from a full deck without replacement. If the first card you draw is a 4,
what is the probability the second card you draw is an ace? The probability of B given that A has already occurred would then mean, drawing an ace on the second card,
given that the first card drawn was a 4. We can break this down. There are 52 cards in a deck, and there are 4 4’s and 4 aces in the deck. Given that the first drawn was a 4,
there are now 51 cards in the deck, and there are still 4 aces, so this would be 4 over 51, which is 0.078 or 7.8%. This can also be shown in a diagram.
This blue box represents our total sample space. Now we add in an orange and yellow circle. Let’s say the orange circle represents the probability of event A
and the yellow circle represents the probability of event B. This means the overlapping area represents the probability of A and B, where both are true.
And there is a formula that is used for conditional probability, the probability of B given that A has already occurred equals the probability of A and B
divided by the probability of A. Using the formula and the diagram, the probability of A and B is this overlapping section here, and the probability of A
is the whole orange circle. Another way to look at it is to focus on what is given, and in this instance it’s given that event A has occurred, so our sample space
has been narrowed to just A. For instance, let’s say that your favorite Baseball team has played 50 games with 28 wins and 22 losses. The yellow circle represents wins,
so 28 total. And the orange circle represents games they scored 5 or more runs in, and let’s say that total is 32. The overlapping area is scored 5 or more runs
and won the game and that total is 20. This means the rest of the area in the yellow circle equals 8, 28 minus 20, and the rest of the orange circle is 12, 32 minus 20.
So, based on this data, what is the probability they win a game given they score 5 or more runs? Since it is given that they score 5 or more runs,
the sample space is narrowed to just the orange circle of 32. And in this orange circle there are 2 sections, wins, the overlapping part of the circle and losses,
the non-overlapping part. The wins area is 20 of the 32 in the new sample space, so 20 divided by 32 equals 0.625 or 62.5%, so, given they score 5 or more runs, there is
a 62.5% probability they win the game. Using the same example, we could use the formula to calculate this. So, the probability of B given that A has already occurred
equals the probability of A and B divided by the probability of A. In this formula, the probability of A and B is the overlapping area the probability they score 5 or
more runs and win, which is 20 out of a total of 50 games, and the probability of A is that they score 5 or more runs, which is 32 out of a total of 50 games.
Now we do the calculations and end up with the same result, 62.5%. Conditional probability can also be used to breakdown data further.
For instance let’s say that you went out and polled 150 people based on 2 things, their age, and whether or not they work out regularly. This table lists those results.
So, let’s find 2 conditional probabilities. First, what is the probability someone works out, given they are in their 40’s, and what is the probability someone is in their 20’s
given they work out regularly? For the first question, it is given that they are in their 40’s, so we need to look at just the data for 40’s in this row here.
And we have 10 respondents that workout, out of a total of 50, so 10 divided by 50 equals 0.2 or 20%. For the second question, it is give that they workout,
so we need to only look at the workout column here. And we have 35 respondents in their 20’s out of a total of 70, so, 35 out of 70 equals 0.5 or 50%.
So, of people who work out regularly, there is a 50% probability they are in their 20’s.
Alright my friends, that be the basics on conditional probability. Hopefully you got something out of this video,
I do have more videos right there for you, till next time, I am outta here.