Basics of Probability

1. Probability Probability&Statistics Study Guides Big Picture Probability is the study of whether an event will or will not occur. Using the laws of probability, we can find the likelihood of two events occurring together, not occurring, or a large variety of other combinations. Key Terms Probability: A measure of how likely an event is. Complementary Event: All the other events in a Event: Something that occurs. Can have one or more sample space. possible outcomes. Compound Event: An event made up of two or more Simple Event: An event that has exactly one outcome. simple events. Sample Space: All the possible outcomes of a Dependent Events: Events whose outcomes do affect experiment. each other. Outcome: The result of a single experiment. Independent Events: Events whose outcomes don’t Experiment: The process of taking a measurement or affect each other. making an observation. Determining Probabilities The probability of an event can be calculated by knowing the number of ways the event can occur and the size of the sample space. • Probability of event A = P(A) = • Probability of an event is always between 0 and 1: 0 = impossible, 1 = always happens The probabilities of all possible outcomes of an event must add up to 1. This means one of the outcomes must happen. This method of determining probabilities assumes that all the possible outcomes are equally likely to happen. Simple Events Example: the probability that a die will land on 3 • Experiment: rolling a single die • Event: the die lands on 3 • This is also a simple event because the die can only land on one number (one possible outcome) • Sample space S lists the possible outcomes: S = {1, 2, 3, 4, 5, 6} • Size of sample space = six • P(die lands on 3) = Non-Simple Events Example: the probability that a die will land on either 2 or 3 • Event: the die lands on 2 or 3 • Can be broken down to two simple events: the die lands on 2, the die lands on 3 • Sample space S = {1, 2, 3, 4, 5, 6} • Size of sample space = six • P(die lands on 2 or 3) = • There are two ways for the event to happen – the die can land on 2 or the die can land on 3 yourtextbookandisforclassroomorindividualuseonly. Disclaimer:thisstudyguidewasnotcreatedtoreplace Complementary Events Complement of an event A = A’ = all the events other than A in the sample space Finding probabilities using complements: • P(A’) = probability that A doesn’t happen Example: Throwing a die • Event A = observing an odd number, Event A’ = observing an even number The Complement Rule: • P(A) + P(A’) = 1 • can be rearranged: P(A’) = 1–P(A) • The Complement Rule is useful when P(A’) is easier to find than P(A) Page 1 of 3 This guide was created by Lizhi Fan and Jin Yu. To learn more about the student authors, visit http://www.ck12.org/about/about-us/team/interns. v1.1.9.2012

2. Probability cont. Compound Probabilities The probability of compound events depend on the outcomes from two or more events. Probability of both A and B happening: P(A and B) = (A B) = P(A) ∙ P(B, given A) • means the intersection of events Probability of either A or B happening: P(A or B) = P(A B) = P(A) + P(B)–P(A B) • means the union of events Probability Conditional Probabilities Conditional probability: the probability of event A occurring, given event B has occurred • Notation: P(A|B), which means “the probability of A, given B” • P(A|B) = • Can rewrite as P(A B) = P(A|B) ∙ P(B) This occurs when events are dependent, where the occurrence of the event will change the probability of the successive events from occurring. • For example, taking a king randomly from a set of cards will decrease the probability of another king being taken out. Independent Events If an event is independent, its occurrence does not change the probability of successive trials from occurring. • For example, rolling a six on a die will not change the probability of rolling a six on the next die. If A and B are independent events, then P(B|A) = P(B). So P(A and B) = P(A) ∙ P(B, given A) = P(A) ∙ P(B) for independent events Mutually Exclusive Events If two events are mutually exclusive, meaning that they cannot occur together, then P(A B) = 0. So P(A or B) = P(A B) = P(A) + P(B) - P(A B) = P(A) + P(B) Page 2 of 3

3. Probability cont. Probability&Statistics Tree Diagrams & Counting Techniques Tree diagrams are useful for showing all the possible outcomes when there is a series of events. Example: A box contains three balls: a red (R) ball, a blue (B) ball, and a white (W) ball. One ball is selected and then returned back into the box. The balls are scrambled and another ball is selected. The tree diagram shows the sample space: • There are 9 possible outcomes. You can also calculate the number of possible outcomes by using the multiplication rule of counting: • If there are n possible outcomes for event A and m for event B, there is n • m possible outcomes for event A followed by event B. Combination and Permutation Counting Rule for Combinations The number of combinations of n objects taken r at a time is: • For example, C6 is the number of combinations of six objects that can be chosen from 13 objects. 13 • Combination = order does not matter To remember that order does not matter for combinations, imagine a combo pizza where you choose whatever toppings you want and just throw it on! Counting Rule for Permutations The number of ways to arrange n objects in order within r positions is: • For example, P6 is the number of ordered ways to arrange six objects chosen from 13 objects. 13 • Permutation = order does matter • For both rules, n and r must be whole numbers, and n ≥ r. Factorial Factorial notation is a shorthand way to write out a common multiplication pattern in probability and statistics: n! = n(n-1)(n-2)(n-3)...(3)(2)(1) • The ! is the symbol for factorial. n! is the n factorial. Law of Large Numbers The law of large numbers simply says that the more an experiment is repeated, the closer the results will get to the theoretical value. For example, say you flip a coin 3 times and get two heads and one tails. The proportion of getting a head seems to be while getting a tail seems like . However, we know that the true value of getting heads or tails should be . If you flip the coin more, the probabilities will even out as the number of trials increase. The probabilities will get closer and closer to chance of getting heads and chance of getting tails. Page 3 of 3