
Probability
Probability deals with the randomness of the universe,
and this inexact science vainly tries to apply numerical understanding
to events that appear to happen in a noncontiguous, nonlinear environment.
What the . . . ?
Oh wait, probability on the SAT. That’s
easy. Probability on the SAT is expressed as a fraction.
The numerator is the number of times a certain event might occur,
and the denominator is the total number of events that might occur.
Here is the probability formula you need to know:
Let’s say you’re playing a game of High Card with your
strange friend Rasputin. High Card is a silly game in which the
two of you randomly pick cards from a 52card deck, and the highest
card wins (now that’s a Friday night of fun!). You are going to
choose from the deck first, and you really want an ace. What’s the
probability that you will draw an ace?
And what’s the probability that you’ll end up with a nonace?
Here’s a more complicated example involving a more exciting
Saturday night with toy cars:

There are five ways to pick a purple toy car because there
are five different purple toy cars. That’s the top number. To find
the bottom number, you need to add up all the different cars, regardless
of color:
5 + 8 + 7 = 20
There are a total of 20 different toy cars to choose from
in all. Therefore, the probability of picking a purple toy car is:
That’s choice A.
When calculating probability, always be sure to
divide by the total number of chances. It would have been tempting
to leave out the number of purple toy cars in the denominator, because
we used them in the numerator, which would have resulted in ,
or , an incorrect answer waiting for
you among the distractors.
Backward Probability
Backward probability is the basic probability item asked
in reverse order. Instead of finding the probability, you are looking
for the total, or real number the probability fraction represents.
For example:

All you need to do is set up the proper proportional equation.
If 1 of 5 toy cars is green and there is a total of 5 green toy
cars, then:
Now we cross multiply the equation to come up with x =
25, choice E.
Probability an Event Won’t Occur
Certain SAT items will ask about the probability of an
event not occurring. No sweat. Figure out the probability
of the event occurring, and subtract that number from 1:
Probability an event will not occur = 1 – probability
an event will occur.
If there is a chance
of rain, then the chance of no rain is ,
or .
Multiple or Unrelated Probabilities
More difficult probability items deal with multiple related
and unrelated events. For these items, the probability of both events
occurring is the product of the outcomes of each event:
A good example of two unrelated events would be: (1) getting
heads on a coin toss and (2) rolling a 5 with a
sixsided die. Just find the probability of each individual event,
and multiply them together. There’s a 1 in 2 chance of getting heads
on a coin and a 1 in 6 chance of rolling a 5. Combining the two
gets you:
The same principle can be applied to find the probability
of a series of events. Let’s keep it simple and stick to toy cars:

To find the probability of Sue drawing 3 green cars
in a row, we need to find the probability of each individual event.
The probability of her drawing a green car on her first try is ,
because there are 9 green cars and 35 total cars (8 blue + 9 green
+ 4 yellow + 14 red = 35 total).
The probability of her drawing a green car on the
second try is slightly different. With one car already removed from
the bag, there are only 34 left, and assuming her first try was
successful, there are only 8 green cars left. So the probability
of drawing a green car the second time is .
Follow the same procedure for the probability of choosing a green
car on the third try, and we come up with .
So the odds of Sue drawing three green cars in a row is:
That’s choice C.
The important point to remember here is that when solving
for the probability of a series of events, always assume
that each prior event was successful, just as we did in
the example above.
Geometric Probability
Another difficult concept the SAT might present is geometric
probability. The same basic concept behind probability still
applies, but instead of dealing with total outcomes and particular
outcomes, you will be dealing with total area and particular area
of a geometric figure. There’s nothing fancy here. Just remember
this formula and you will be fine:
