Probability
Probability
The probability is high that at least one question on the SAT will cover probability. The probability is even higher that the probability formula will help you on SAT probability questions. Here’s the formula:
Let’s say you go on a game show and are put in a room that contains 52 doors. Behind 13 of the doors are awesome prizes, including new cars, diamond watches, and infinity millions of dollars. Behind the rest of the doors are goats. What’s the probability that you’ll draw an awesome prize?
And what’s the probability that you’ll end up with the goat?
Here’s a more complicated example, involving the SAT’s favorite probability prop: marbles! That SAT sure knows how to have a good time with marbles.
Joe has 3 green marbles, 2 red marbles, and 5 blue marbles, and if all the marbles are dropped into a dark bag, what is the probability that Joe will pick out a green marble?
There are three ways for Joe to pick a green marble (since there are three different green marbles), but there are ten total possible outcomes (one for each marble in the bag). Therefore, the probability of picking a green marble is
When you calculate probability, always be careful to divide by the total number of chances. In the last example, you may have been tempted to leave out the three chances of picking a green marble from the total possibilities, yielding P = 3/7. Brutal wrongness.
Backwards Probability
The SAT might also ask you a “backwards” probability question. For example, if you have a bag holding twenty marbles, and you have a 1/5 chance of picking a blue marble, how many blue marbles are in the bag? All you have to do is set up the proper equation, following the model of P = m /n:
in which x is the variable denoting the number of blue marbles. Cross-multiplying through the equation, you get 5x = 20., which reduces to x = 4.
The Range of Probability
The probability, P, that any event will occur is always 0 ≤ P ≤ 1. A probability of 0 for an event means that the event will never happen. A probability of 1 means the event will always occur. Drawing a bouquet of flowers from a standard deck of cards has a probability of 0. Becoming Lord (or Lady) of the Universe after scoring 2400 on the new SAT has a probability of 1.
The Probability That an Event Will Not Occur
Some SAT questions ask you to determine the probability that an event will not occur. In that case, just figure out the probability of the event occurring, and subtract that number from 1.
Probability and Multiple Unrelated Events
More difficult SAT probability questions deal with multiple unrelated events. For these questions, the probability of both events occurring is the product of the outcomes of each event: , where is the probability of the first event, and is the probability of the second event.
A perfect example of two unrelated events is this: Drawing a spade from a full deck of cards and rolling a one with a six-sided die is the product of the probability of each event. Neither outcome will affect the outcome of the other. The probability of both events occuring is
The same principle can be applied to finding the probability of a series of events. Take a look at the following problem:
A teacher keeps a jar full of different flavored jelly beans on her desk and hands them out randomly to her class. But one greedy student likes only the licorice-flavored ones. One day after school, the student sneaks into the dark classroom and steals three jelly beans. If the jar has 50 beans in all—15 licorice, 10 cherry, 20 watermelon, and 5 blueberry—what is the probability that the student got at least one licorice-flavored bean?
In order to find the probability of three consecutive events, first find the probability of each event separately. The first jelly bean has a 15 /50 chance of being licorice-flavored. The second jellybean, however, is a different story. There are now only 49 jelly beans left in the jar, so the probability of getting another licorice-flavored one is 14/49. The probability of getting a third licorice-flavored jelly bean is 13/48. The odds of all three happening are:
The moral of this sad tale of larceny and candy is that crime pays only 13/ 560 of the time.
Geometric Probability
The new SAT occasionally asks questions to which it has given the exciting name “geometric probability.” The SAT could have saved itself some time by just saying that it’s going to ask you questions about playing darts.
What is the probability of throwing a dart into the shaded area of the dartboard pictured above?
Here you have to find the area of some shaded (or unshaded) region, and divide that by the total area of the figure. In this question, the dartboard is a circle of radius 3. The shaded region is the area of the circle minus a circle of radius 2.
and 0.56 equals 56%.
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