PROBABILITY

Counting methods can be used to find the number of possible ways to choose objects with and without regard to order. The Fundamental Counting Principle: If there are “a” ways for one activity to occur, and “b” ways for a second activity to occur, then there are a  b ways for both to occur. Example: How many different ice-cream sundaes can be made if you have four flavors of ice-cream, three different toppings, and two different sauces? 4 x 3 x 2 = 24 different sundaes

A PERMUTATION is an arrangement of objects in a specific order. The order of the arrangement is important! The notation for a permutation: n P r Where n is the total number of objects and r is the number of objects chosen Example: How many different ways can you arrange five books on a shelf if you have seven books from which to choose? n = 7 and r = 5  7 P 5 = 2520 ways

FACTORIAL If n = r, then the permutation is considered to be a “factorial”. The formula n P r = n! Example: How many different ways can you arrange all six books on a shelf? Let n = 6  n! = 6! 6 x 5 x 4 x 3 x 2 x 1 = 6! or 720 different ways

A COMBINATION is an arrangement of objects without regard to order. The order of objects is not important. The notation for a combination: n C r Where n is the total number of objects and r is the number of objects chosen Example: You have twenty students in your homeroom. How many ways can you choose three students to represent your homeroom for student council? n = 20, r = 3 (the order does not matter) 20 C 3 = 1140 different ways

An outcome is the result of a single trial, such as tossing a die. The sample space is all the possible outcomes. An event is any outcome or group of outcomes. The outcomes that match a given event are favorable outcomes. TERMINOLOGY

Event: rolling an even number Sample Space:{1, 2, 3, 4, 5, 6} Favorable Outcomes:{2, 4, 6}

There are two kinds of probability: Theoretical and Experimental (Empirical) Although both of these methods for determining the probability of an event involve the concept of "chance", the solutions are obtained in two very different ways.

EXPERIMENTAL PROBABILITY is the likelihood a specific outcome will occur, based on repeated testing and observation of results as in an experiment. It is an estimate that the event will happen based on how often the event occurs after collecting data or running an experiment with a large number of trials. It is the ratio of the number of times an event occurs to the number of trials.

Experimental probability is calculated after the event has occurred. By observing the pattern of events and how often a certain outcome has occurred, we try to estimate how often we can expect to see a certain outcome in the future. The more data that can be analyzed, the more accurate your estimate is likely to be. In other words, the experiment should have many trials.

Example: To determine the experimental probability of tossing “tails” on a coin, toss the coin about 100 times. Then divide the number of times you tossed “tails” by 100. Your probability may not be ½ as it would be with theoretical probability.

Experimental Probability On a production line, 45 toasters were found to be defective out of the 1500 randomly selected toasters that were tested. What is the probability that a toaster is defective? P(defective) = 45 / 1500 = 3 / 100

THEORETICAL PROBABILITY is the likelihood a specific outcome will occur, determined by calculating expected outcomes under ideal circumstances. It is the ratio of the number of favorable outcomes to the number of possible outcomes.

Theoretical probability is calculated before any event has taken place. Example: The probability of rolling a “5” on a die is 1 / 6.

Theoretical Probability A jar contains 7 red marbles, 3 green marbles, 6 yellow marbles, and 4 blue marbles. If a single marble is chosen from the jar, what is the probability it will be red? Green? Yellow? Blue? P(red) = 7 / 20 P(green) = 3 / 20 P(yellow) = 6 / 20 = 3 / 10 P(blue) = 4 / 20 = 1 / 5

COMPLEMENT OF AN EVENT. The complement of an event is its “opposite,” or the event NOT happening. If you state an event with the symbol A, the complement of the event is written as a A or A’. The sum of the probabilities of an event and its complement is always equal to one. P(A) + P(A’) = 1 The probability of the complement of an event is equal to one minus the probability of the event: P(A’) = 1 - P(A)

Example: If you select one card at random from a standard deck of cards, what is the probability that the card is not a queen? P(queen) = 4 / 52 or 1 / 13  P(not a queen) = 1 – 1 / / 13 – 1 / / 13 P(Q) = 1 / 13 P(Q’) = 1 – 1 / 13 = 12 / 13

The probability of an event A, is a number between 0 and 1, inclusive. If P(A) > P(B) then event A is more likely to occur than event B If P(A) = P(B) then events A and B are equally likely to occur If event A is impossible, then P(A) = 0 If event A is certain, then P(A) = 1 The complement of event A is A’: P(A’) = 1 – P(A)

A jar of marbles has 5 green, 3 pink, 6 blue, and 4 brown. What is the probability that you select a marble at random and it is purple? P(purple) = 0 (an impossible event) A bag contains certain numbers written on pieces of paper. The numbers are {0, 2, 4, 6, 8, 10, 12}. If you randomly pick a number from the bag, what is the probability that the number is an even number? P(even) = 1 (a certain event)

COMPOUND EVENTS Compound events or combined events involve putting together two or more events. The probability of two compound events is found by adding the probability of each event. When two events cannot happen at the same time, they are mutually exclusive events. The probability is determined by adding the probabilities of each event. Mutually Exclusive Events P(A or B) = P(A) + P(B)

Mutually Exclusive Example: A jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that the marble is yellow or green? P(yellow or green) = P(yellow) + P(green) = 4 / / 10 = 7 / 10

When two events can occur at the same time, they are inclusive or overlapping. For any two events which are not mutually exclusive, the probability that an outcome will be in one event or the other event is the sum of their individual probabilities minus the probability of the outcome being in both events. Inclusive Events (Events that CAN happen at the same time) P(A or B) = P(A) + P(B) - P(A and B)

Example: Find the probability of selecting a queen or a red card from a standard deck of cards. These events are inclusive events because you can select two cards that are both red and a queen: queen of hearts and queen of diamonds. P(queen) = 4 / 52 P(red) = 26 / 52 P(queen and red) = 2 / 52 P(queen or red) = P(queen) + P(red) - P(queen and red) = ( 4 / / 52 ) – 2 / 52 = 28 /52 = 7 / 13

INDEPENDENT EVENTS Two events are said to be independent if the result of the second event is not affected by the result of the first event. Probability of Two Independent Events P(A and B) = P(A)  P(B) Key Words: replaced, returned, put back, with replacement

Example: What is the probability that you pick a king from a deck of cards, replace the card, then pick another king? P(king and king) = 4 / 52  4 / 52 = 1 / 13  1 / 13 = 1 / 169

DEPENDENT EVENTS Two events are dependent events if the occurrence of one event affects the probability of the second event. Probability of Two Independent Events P(A)  P(B after A) Key Words: without replacing, keep it, do not return, without replacement

Example: What is the probability that you pick a king from a deck of cards, don’t replace the card, then pick another king? P(king then king) = P(king)  P(king after the 1 st king) = 4 / 52  3 / 51 = 1 / 13  1 / 17 = 1 / 221