Statistics Essentials For Dummies (48 page)

The formula for the conditional probability of
A
given
B
is

P(
A
|
B
) =
. You're dividing by P(
B
) because you know

event
B
has happened, making this the new sample space. But because the denominators of P(
A
B
) and P(
B
) both equal the grand total in the two-way table, you can find the conditional probability by just taking the number in the cell representing
A
B
, divided by the appropriate row or column total for event
B
. (The denominators for these probabilities are the same, the grand total, and they cancel out when you divide the probabilities, so you don't have to include them in the calculations.)

For example, look at the event that the player makes the second free throw given he made the first one. This event is denoted
Y
₂
|
Y
₁
. (
Y
₁
is the event that is known, so you use its marginal total in the denominator.) So the probability that a player makes the second free throw given he made the first one, P(
Y
₂
|
Y
₁
), is found by taking the number in the cell representing
Y
₂
Y
₁
, 60, and divide by the row total representing
Y
₁
, 100. So the probability of making the second shot, given he made the first, is 0.60 or 60%.

Now look at the event that the player misses the second free throw given he missed the first one. This event is denoted
N
₂
|
N
₁
. So the probability that a player misses the second free throw given he missed the first one, P(
N
₂
|
N
₁
), is found by taking the number in the cell for
N
₂
N
₁
, 10, and dividing by the second row total, 55, to get 0.18 or 18%.

The general formula for finding the conditional probability of an event in row
i
given an event in row
j
of the two-way table

is P(row
i
|column
j
) =
. The general formula

for finding the conditional probability of the event in column
j
given the event in row
i
of the two-way table is

P(column
j
|row
i
) =
.

A conditional probability is the probability of one event happening given another event is known to have occurred. Using a two-way table, to find the conditional probability of an event in a certain column of the table given an event in a row of the table, take the cell count for the intersection divided by the corresponding row total. To find the conditional probability of an event in a certain row of the table given an event in a column of the table, take the cell count for the intersection divided by the corresponding column total. The clue that it's a conditional probability is the fact that you know one event is known to have occurred. Words like
given
,
knowing
, and
of
are often used to mean conditional probability.

Here is a summary of the conditional probabilities we can calculate regarding the free throws example:

The probability of making the second shot given he made the first one, P(
Y
₂
|
Y
₁
), is 60/100 = 0.60 or 60%.