Algebra Review - Probability

NOTE: This review was borrowed from SparkNotes

Simple Probability
1.1 Probability

Probability is a measure of the likelihood that an event will happen.

When dealing with probability, the outcomes of a process are the possible results. For example, when a die is rolled, the possible outcomes are 1 , 2 , 3 , 4 , 5 , and 6 . In mathematical language, an event is a set of outcomes, which describe what outcomes correspond to the "event" happening. For instance, "rolling an even number" is an event that corresponds to the set of outcomes {2,4,6} . The probability of an event, like rolling an even number, is the number of outcomes that constitute the event divided by the total number of possible outcomes. We call the outcomes in an event its "favorable outcomes".

 Probability = # favorable outcomes # possible outcomes

If a die is rolled once, determine the probability of rolling a 4 : Rolling a 4 is an event with 1 favorable outcome (a roll of 4 ) and the total number of possible outcomes is 6 (a roll of 1 , 2 , 3 , 4 , 5 , or 6 ). Thus, the probability of rolling a 4 is [1/6] .
If a die is rolled once, determine the probability of rolling at least a 4: Rolling at least 4 is an event with 3 favorable outcomes (a roll of 4 , 5 , or 6 ) and the total number of possible outcomes is again 6 . Thus, the probability of rolling at least a 4 is [3/6] = [1/2] .

Here are two more examples:
If a coin is flipped twice, determine the probability that it will land heads both times:

Favorable outcomes: 1 -- HH
Possible outcomes: 4 -- HH, HT, TH, TT

Thus, the probability that the coin will land heads both times is [1/4] .

If Dan grabs one sock from a drawer containing 3 white socks, 4 blue socks, and 5 yellow socks, what is the probability that he will grab a white sock?

Favorable outcomes: 3 (3 white socks)
Possible outcomes: 12 (3 white socks + 4 blue socks + 5 yellow socks)
Thus, the probability that Dan will grab a white sock is [3/12] = [1/4] .

Though probabilities are calculated as fractions, they can be converted to decimals or percents--the Fractions SparkNote in Pre-Algebra explains how to convert fractions to decimals and the SparkNote on Percents describes how to convert them to percents.

1.2 Boundaries on Probability
If all outcomes are favorable for a certain event, its probability is 1 . For example, the probability of rolling a 6 or lower on one die is [6/6] = 1 .

If none of the possible outcomes are favorable for a certain event (a favorable outcome is impossible), the probability is 0 . For example, the probability of rolling a 7 on one die is [0/6] = 0 .

All probabilities are between 0 and 1 : 0 £ p £ 1

Complementary Events and Odds
2.1 Complementary Events

Two events are said to be complementary when one event occurs if and only if the other does not. The probabilities of two complimentary events add up to 1 .

For example, rolling a 5 or greater and rolling a 4 or less on a die are complementary events, because a roll is 5 or greater if and only if it is not 4 or less. The probability of rolling a 5 or greater is [2/6] = [1/3] , and the probability of rolling a 4 or less is [4/6] = [2/3] . Thus, the total of their probabilities is [1/3] + [2/3] = [3/3] = 1 .

Example: If the probability of an event is [3/8] , what is the probability of its complement?

The probability of its complement is 1 - [3/8] = [8/8] - [3/8] = [5/8] .

2.2 Odds

The odds of an event is the ratio of the probability of an event to the probability of its complement. In other words, it is the ratio of favorable outcomes to unfavorable outcomes. We say the odds are "3 to 2," which means 3 favorable outcomes to every 2 unfavorable outcomes, and we write 3:2 . For example, the odds of rolling a 5 or greater are 2:4 , which reduces to 1:2 .

Example 1: If we flip a coin two times, what are the odds for it landing heads at least once?

Favorable outcomes: 3 -- HH, HT, TH.
Unfavorable outcomes: 1 -- TT.

Thus, the odds for it landing heads at least once are 3 to 1 , or 3:1 .

Example 2: If the probability of an event happening is [2/7] , what are the odds for that event?

Since the probability of the event is [2/7] , the probability of its complement is 1 - [2/7] = [7/7] - [2/7] = [5/7] . Thus, the odds for that event are [2/7]:[5/7] , which is equivalent to 2:5 .

Example 3. If the odds for an event are 3:2 , what is the probability of the event happening?

Favorable outcomes = 3 .

Possible outcomes = favorable outcomes + unfavorable outcomes = 3 + 2 = 5 .
Thus, the probability of the event happening is [3/5] .

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