Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

Chapter 15 Chances, Probabilities, and Odds Measuring Uncertainty

Chances, Probabilities, and Odds Outline/learning Objectives To describe an appropriate sample space of a random experiment. To apply the multiplication rule, permutations, and combinations to counting problems. To understand the concept of a probability assignment.

Chances, Probabilities, and Odds Outline/learning Objectives To identify independent events and their properties. To use the language of odds in describing probabilities of events.

Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces

Chances, Probabilities, and Odds Random experiment Description of an activity or process whose outcome cannot be predicted ahead of time. Sample space Associated with every random experiment is the set of all of its possible outcomes. We will consistently use the letter S to denote a sample space and N to denote its size (the number of outcomes in S).

Chances, Probabilities, and Odds Rolling the Dice: Part 1 One of the most common things we do with dice is to roll a pair of dice and consider just the total of the the two die. A more general scenario is when we do care what number each individual turns up. Here below we have a sample space with 36 different outcomes.

Chances, Probabilities, and Odds Rolling the Dice: Part 1 When looking at the figure below you will notice that we are treating the dice as distinguishable objects (as if one were white and the other red), so that and are considered different outcomes.

Chances, Probabilities, and Odds 15.2 Counting Sample Spaces

Chances, Probabilities, and Odds The Multiplication Rule When something is done in stages, the number of ways it can be done is found by multiplying the number of ways each of the stages can be done.

Chances, Probabilities, and Odds The Making of a Wardrobe: Part 2 Our strategy will be to think of an outfit as being put together in stages and to draw a box for each of the stages. We then separately count the number of choices at each stage and enter that number in the corresponding box.

Chances, Probabilities, and Odds The Making of a Wardrobe: Part 2 The last step is to multiply the numbers in each box. The final count for the number of different outfits is N = 3  7  27  3 = 1701

Chances, Probabilities, and Odds 15.3 Permutations and Combinations

Chances, Probabilities, and Odds Permutation A group of objects where the ordering of the objects within the group makes a difference. Combination A group of objects in which the ordering of the objects is irrelevant.

Chances, Probabilities, and Odds The Pleasures of Ice Cream: Part 1 Say you want a true double in a bowl – how many different choices so you have?

Chances, Probabilities, and Odds The Pleasures of Ice Cream: Part 1 The natural impulse is to count the number of choices using the multiplication rule (and a box model) as shown below. This would give an answer of 930.

Chances, Probabilities, and Odds The Pleasures of Ice Cream: Part 1 Unfortunately, this answer is double counting each of the true doubles. Why?

Chances, Probabilities, and Odds The Pleasures of Ice Cream: Part 1 When we use the multiplication rule, there is a well-defined order to things, and a scoop of strawberry followed by a scoop of chocolate is counted separate from a scoop of chocolate followed by a scoop of strawberry. The good news is that now we understand why the count of 930 is wrong and we can fix it. All we have to do is divide the original count by 2. (31  30)/2 = 465

Chances, Probabilities, and Odds 15.4 Probability Spaces

Chances, Probabilities, and Odds Event Any subset of the sample space. Simple event An event that consists of just one outcome. Impossible event A special case of the empty set { }, corresponding to an event with no outcomes.

Chances, Probabilities, and Odds Probability assignment A function that assigns to each event E a number between 0 and 1, which represents the probability of the event E and which we denote by Pr (E). Probability space Once a specific probability assignment is made on a sample space, the combination of the sample space and the probability assignment.

Chances, Probabilities, and Odds Elements of a Probability Space Sample space: S = {o1, o2,…., oN} Probability assignment: Pr(o1),Pr(o2),… Pr(oN) [Each of these is a number between 0 and 1 satisfying Pr(o1) + Pr(o2) + … Pr(oN) = 1] Events: These are all the subsets of S, including { } and S itself. The probability of an event is given by the sum of the probabilities of the individual outcomes that make up the event. [In particular, Pr({ }) = 0 and Pr(S) =1]

Chances, Probabilities, and Odds 15.5 Equiprobable Spaces

Chances, Probabilities, and Odds Probabilities in Equiprobable Spaces Pr(E) = k/N (where k denotes the size of the event E and N denotes the size of the sample space S). A probability space where each simple event has an equal probability is called an equiprobable “equal opportunity” space.

Chances, Probabilities, and Odds Rolling the Dice: Part 2 The sample space has N = 36 individual outcomes, each with probability 1/36. We will use the notation T2, T3, …T12 to describe the events “roll a total of 2,” “roll a total of 3,” …, “roll a total of 12,” respectively. We show you how to find Pr(T7) and Pr(T11), T11 = , Thus, Pr(T11) = 2/36  0.056 T7 = , Thus, Pr(T7) = 6/36 = 1/6  0.167

Chances, Probabilities, and Odds Tallying We can just write down all the individual outcomes in the event E and tally their number. This approach gives and Pr(E) = 11/36.

Chances, Probabilities, and Odds Complementary Event Imagine that you are playing a game, and you win if at least one of the two numbers comes up an Ace (that’s event E). Otherwise you lose (call that event F). The two events E and F are called complementary events. The probabilities of complementary events add up to 1. Thus, Pr(E) = 1 – Pr(F).

Chances, Probabilities, and Odds Independence Events If the occurrence of one event does not affect the probability of the occurrence of the the other. Multiplication Principle for Independent Events When events E and F are independent, the probability that both occur is the product of their respective probabilities; in other words, Pr (E and F) = Pr(E) • Pr(F).

Chances, Probabilities, and Odds

Chances, Probabilities, and Odds Let E be an arbitrary event. If F denotes the number of ways that event E can occur (the favorable outcomes or hits), and U denotes the number of ways that event E does not occur (the unfavorable outcomes, or misses), then the odds of (also called the odds in favor of), the event E are given by the ratio F to U, and the odds against the event E are given by the ratio U to F.

Chances, Probabilities, and Odds Conclusion Sample space Random experiment Events Probability assignment Equiprobable spaces