Exercise 2 Page 680

We want to complete the given sentence.

The probability that event $B$ will occur given that event $A$ has occurred is called the ___________ of $B$ given $A$ and is written as ______.

Consider that

A

and

B

are dependent events. This means that the occurrence of

A

affects the occurrence of

B .

This changes the probability of

B

when

A

occurs. We write this probability as follows.

P (B ∣ A)

This is called the probability of

B

given

A .

The probability that event $B$ will occur given that event $A$ has occurred is called the probability of $B$ given $A$ and is written as $\underline{P (B ∣ A)} .$

Let's illustrate this with an example!

Example

Consider that we draw a card from a deck of cards and then, without replacement, we draw another card from the deck. Suppose that we want the probability of drawing a $3$ of hearts and then a $5$ of spades.

We know that there are

52

cards in a deck. We can find the probability of drawing a

3

of hearts.

P (3 of hearts) = \frac{1}{5 2}

Since there is no replacement, there are

51

cards left in the deck. Now we can calculate the probability of drawing a

5

of spades, given that we have already drawn a

3

of hearts.

P (5 of spades ∣ 3 of hearts) = \frac{1}{5 1}

We can see that the event of drawing a

3

of cards affected the probability of drawing a

5

of spades.