Statistics Essentials For Dummies (49 page)

The probability of missing the second shot given he made the first one, P(
N
₂
|
Y
₁
), is 40/100 = 0.40 or 40%. Observe that this is 1 - 0.60, because you either miss the second or you don't. (That is, making the shot and missing the shot are complements of each other.)

The probability of missing the second shot given he missed the first one, P(
N
₂
|
N
₁
), is 10/55 = 0.18 or 18%.

The probability of making the second shot given he missed the first one, P(
Y
₂
|
N
₁
), is 45/55 = 0.82 or 82%. Note this is the complement of the previous event, so their probabilities sum to one.

Once you're given that event
B
has happened,
A
either happens or it doesn't. So it is true that P(
A
|
B
) + P(
Ac
|
B
) = 1. But it is
not
true that P(
A
|
B
) + P(
A
|
Bc
) = 1 because in each term you are conditioning on a different event. This is a common mistake that you definitely want to avoid. (The notation statisticians use for the event where A doesn't happen is Ac. We call it "A complement.")

Checking for Independence

You know he makes the second free throw more often than the first, from the section on marginal probabilities. Now you are ready to answer your second question: In situations where he misses the first shot, does he make the second shot even more often? If the answer is yes, then we say the outcome of the second shot is related to, or is
dependent
, on, the outcome of the first shot. If the answer is no, then we say the outcome of the second shot is not related to, or is
independent
of, the outcome of the first.

Formally speaking, two events
A
and
B
are
independent
if P(
A
|
B
) = P(
A
). In other words, if the knowledge that
B
has happened does not change the probability of
A
happening, then events
A
and
B
are independent. Note this also means that if events
A
and
B
are independent, then P(
A
|
B
) = P(
A
|
Bc
), because both of these terms must be equal to P(
A
) in that case.

To summarize these results, you can show events
A
and
B
are independent using two different methods:

Method 1: If P(
A
|
B
) = P(
A
) then
A
and
B
are
independent
. If they are not equal, then we say
A
and
B
are
dependent
.

Method 2: If P(
A
|
B
) = P(
A
|
Bc
) then
A
and
B
are
independent
. If they are not equal, then we say
A
and
B
are
dependent
.

You only have to check for independence using one of those methods; you need not use both.

Using method 1 to answer your question, you check for independence by comparing his overall rate of making the second shot to his rate of making the second shot when you know he's missed the first. That is, check to see if P(
Y
₂
) = P(
Y
₂
|
N
₁
). You know that P(
Y
₂
) is the overall chance of making the second shot, which equals 105/155 = 0.68 or 68%. Now if the first shot was missed, the probability of making the second shot increases to P(
Y
₂
|
N
₁
) = 45/55 = 0.82 or 82%. Because 0.68 is not equal to 0.82, the outcomes of the two shots are dependent. In situations where the first free throw is missed, he makes the second one more often than his overall rate.

Using method 2, you check to see if the probability of making the second shot is the same whether the first shot is made or missed. That is, check to see if P(
Y
₂
|
Y
₁
) = P(
Y
₂
|
N
₁
). When the first shot is made, the chance of making the second is P(
Y
₂
|
Y
₁
) = 60/100 = 0.60 or 60%; when the first shot is missed, the chance of making the second shot increases to P(
Y
₂
|
N
₁
) =
45/55 = 0.82 or 82%. Since these probabilities are not equal, the outcomes of the two shots are dependent. The probability of making the second shot is higher when he misses the first one than when he makes the first one.