All language subtitles for 010 The Multiplication Law_en

-: Hello again.

In this video we will examine the multiplication rule.

For starters, let us examine

the conditional probability formula once again.

The probability of event A, given B

equals the probability of the intersection

of A and B over the probability

of event B occurring.

We can multiply both sides of the equation by

P of B to get the probability of the intersection

of A and B equals the probability

of A, given B times the probability of B.

We call this equation the multiplication rule.

Let us look at a numerical example

and see why this makes sense.

Suppose the probability of event B is 0.5

and the probability of event A given B is 0.8.

This suggests that event B occurs 50% of the time

and event A also appears in 80% of those

50% when B occurred.

Therefore, the likelihood of A and B

occurring simultaneously is 0.8 x 0.5, or 0.4.

Let us look at another example.

Suppose we draw two cards

from a standard deck of 52 playing cards.

We draw one,

shuffle the deck without returning the card

and then draw a second one.

What is the probability

of drawing a spade on the second draw

and not drawing a spade on the first draw?

If we express these as a single conditional probability

event A would be drawing a spade on the second try,

and event B would be not drawing a spade on the first try.

As stated before, the likelihood

of drawing a specific suit is one fourth or 0.25.

We already discussed how to calculate

the probability of complements,

so the probability of B would equal 1-0.25 or 0.75.

Now, be careful when estimating the probability

of drawing a spade on the second turn.

There are only 51 cards left,

so we must adjust the favorable overall formula

to find the new likelihood.

We have assumed we did not draw spade on the first go,

so the favorable outcomes would still be the 13 spades left.

However, we are one card short from having a complete deck,

so the new sample space would be 51; therefore,

the probability would be 13 over 51 or approximately 0.255.

Great. So far we have calculated the likelihood

of not drawing a spade on the first turn,

and the probability of drawing a spade

on the second go given we drew something else first.

However, we still haven't

answered the question we are interested in.

What is the probability

of drawing a spade on the second draw

and not drawing a spade on the first draw?

To answer it, we need to apply the multiplication rule.

We plug in 0.255 for P of A given B

and 0.75 for P of B in the multiplication law formula

to get a probability of close to 0.191

for the intersection of A and B.

We have a probability of 0.191

of drawing a spade on the second turn,

assuming we did not draw one initially.

Marvelous work everyone!

For homework you can practice your understanding

by finding the probability of getting a spade

on either the first turn or the second turn.

You could solve it in two separate ways,

by using the additive law,

or by using the multiplicative rule.

Thanks for watching.