


Tackling No Problem with Probability
As confusing and frustrating as these items can get, they
can be easily managed if you learn to employ the threestep method
described below:
Step 1: OVER—Identify the number of outcomes that
fit the item’s requirements.
Step 2: UNDER—Identify the total number of outcomes
possible.
Step 3: Place the OVER over the UNDER. Give the item
what it wants.
No Problem with Probability in Slow Motion
Now let’s look at the steps more closely, starting with
a fairly common probability item.

Step 1: OVER—Identify the number of outcomes that fit
the item’s requirements.
Be careful: you don’t want to jump in too fast and say
3. In an ordinary deck of playing cards, each card appears one time
for each of the four suits. Therefore, the total number of outcomes
that fit the requirements is 12, four suits for each of the three
face cards.
Step 2: UNDER—Identify the total number of outcomes possible.
If you are pulling a card at random from a 52card deck,
then there are 52 total outcomes possible.
Step 3: Place the OVER over the UNDER. Give the item what
it wants.
Create a fraction with the numbers you came up with
for OVER and UNDER. , answer B.
Guided Practice
Try this one on your own.

Step 1: OVER—Identify the number of outcomes that fit
the item’s requirements.
Is there more than one probability here?
Step 2: Under—Identify the total number of outcomes possible.
Think carefully about all the possibilities.
Step 3: Place the OVER over the UNDER. Give the item what
it wants.
Do you remember what to do to find the probability of
multiple events?
Guided Practice: Explanation
This is a variation on the standard probability item.
You still want to follow the basic steps, but notice how they slightly
differ.
Step 1: OVER—Identify the number of outcomes that fit
the item’s requirements.
Because the item asks about multiple events, we need to
find the probability of each individual event first. The first time
a piece of candy is picked from the bag, how many are chocolate?
8. Hold on to this number, we’ll come back to it. The second time
a piece of candy is picked from the bag how many pieces of chocolate
will there be (always assume the previous trials were successful)?
7. That’s it for step 1.
Step 2: UNDER—Identify the total number of outcomes possible.
This is the exact same thing we did in step 1, but now
we are concerned only with the total pieces of candy. The first
time a piece of candy is picked there is a total of 16 pieces, but
the second time the total changes to 15. Keep these numbers handy.
Step 3: Place the OVER over the UNDER. Give the item what
it wants.
Combine the numbers you came up with to create two
probabilities. The probability of getting a piece of chocolate the
first time is . The probability of getting a piece
of chocolate the second time would be . To
find the probability of these two events happening consecutively—in a
row—just multiply them together:
Answer E is correct.
Independent Practice
Once you’ve completed this item, turn the page for the
solution.

Independent Practice: Explanation
This item is one of the more difficult probability items
you’re likely to encounter on the SAT. But you still want to stick
with our trusty threestep method.
Step 1: OVER—Identify the number of outcomes that fit
the item’s requirements.
All right, the stem asks for heads coming up at
least once, so this means we need to include the probability
of getting one head, two heads, and all three heads, as well as
any of the different orders in which they can appear in three tosses.
That’s a lot of thinking, and with the time constraints of the test,
you’re not going to have enough time to write out all the different
outcomes where you get at least one head.
Let’s try approaching this item with a little creativity.
The stem asks for at least one head, but consider
the opposite of this stem—What would be the probability of NO heads?—which
is the same as asking for the probability of all tails. Looking
at the item this way, we can identify the number of outcomes in
which we get all tails. There is one chance on the first toss, one
chance on the second toss, and one chance on the third toss.
Step 2: UNDER—Identify the total number of outcomes possible.
Whenever we toss a coin, there are only two possible outcomes.
Because all three tosses are independent of one other—that is, the
coin is picked up and tossed again—our total outcomes for each individual
toss is 2.
Step 3: Place the OVER over the UNDER. Give the item what
it wants.
To determine the probability of multiple events,
you need to multiply the probability of each individual event together.
So the probability of getting tails on all three tosses—which is
the same as never getting heads—would be .
Don’t be fooled by answer choice B, .
The item did not ask for the probability of tossing tails three
times. The item asks for the probability of tossing at least one
head, which is the same as not tossing three tails.
Now we subtract the probability of three tails from 1, and the result
is the probability of tossing at least one head. answer D.
