Probability is another related topic you might see on the Math IIC. You should be familiar with the probability formula and with applying the probability formula to calculate the likely outcome of independent events.
The Probability Formula
The probability of an event is a number between 0 and 1 that represents the likelihood of that event occurring. You can calculate the probability of an event by dividing the number of desired outcomes by the total number of possible outcomes.
For example, in a deck of 52 cards, the probability of pulling one of the 13 hearts from the deck is much higher than the likelihood of pulling out the ace of spades. To calculate an exact value for the probability of drawing a heart from the deck, divide the number of hearts you could possibly draw by the total number of cards in the deck.
In contrast, the possibility of drawing the single ace of spades from the deck is:
After looking at these examples, you should be able to understand the general formula for calculating probability. Let’s look at a more complicated example:
Joe has 3 green marbles, 2 red marbles, and 5 blue marbles. If all the marbles are dropped into a dark bag, what is the probability that Joe will pick out a green marble?
There are 3 ways for Joe to pick a green marble (since there are 3 different green marbles), but there are 10 total possible outcomes (one for each marble in the bag). Therefore, you can simply calculate the probability of picking a green marble:
When calculating probabilities, always be careful to count all of the possible favorable outcomes among the total of possible outcomes. 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 the equation P = 3 /7. That would have been wrong.
The Range of Probability
The probability, P, of any event occurring will always be 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. For example, drawing a green card from a standard deck of cards has a probability of 0; getting a number less than seven on a single roll of one die has a probability of 1.
If you are ever asked a probability question on the Math IIC, you can automatically eliminate any answer choices that are less than 0 or greater than 1.
The Probability That an Event Will Not Occur
Some Math IIC 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 Events
The most difficult Math IIC probability questions deal with the probability of multiple events occurring. Such questions will always deal with independent events— events whose probability is not dependent on the outcome of any other event. For these questions, the probability of both events occurring is the product of the outcomes of each event: P(A) P(B), where P(A) is the probability of the first event and P(B) is the probability of the second event.
For example, the probability of 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.
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 particularly picky student likes only the licorice-flavored ones. If the jar has 50 beans in all—15 licorice, 10 cherry, 20 watermelon, and 5 blueberry— what is the probability that the first three jelly beans picked out are licorice flavored?
In order to find the probability of three consecutive events, you should first find the probability of each event separately. The first jelly bean has a 15 /50 chance of being licorice-flavored. The second jelly bean, 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 third jelly bean has a probability of 13 /48. The odds of all three happening is:
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