Chapter 6 6.1/6.2 Probability Probability is the branch of mathematics that describes the pattern of chance outcomes

The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. Definition: The probability of any outcome of a chance process is a number between 0 (never occurs) and 1(always occurs) that describes the proportion of times the outcome would occur in a very long series of repetitions. Probability is long-term relative frequency

Thinking about randomness You must have a long series of independent trials The outcome of one trial must not influence the outcome of any other Empirical Probability – the probability of simulations Theoretical Probability – the probability of mathematical formulas Computer simulations are very useful because we need long runs of trials Short runs give only rough estimates of a probability

A game of chance Page 334 #6.2 In the game of heads or Tails, Betty and Bob toss a coin four times. Betty wins a dollar from Bob for each head and pays Bob a dollar for each tail- that is, she wins or loses the difference between the number of heads and the number of tails. For example, if there are one head and three tails, Betty loses $2. You can check that Betty’s possible outcomes are {-4, -2, 0, 2, 4} Assign probabilities to these outcomes by playing the game 20 times and using the proportions for the outcomes as estimates of the probabilities.

6.2 Probability Models Sample space: set of all possible outcomes Probability model: a mathematical description of a random phenomenon consisting of two parts: a sample space and a way of assigning probabilities What is the sample space of rolling two die? Display the outcomes How many outcomes? What are the possible sums of the die?

Example: Roll the Dice Give a probability model for the chance process of rolling two fair, six-sided dice – one that’s red and one that’s green. Sample Space 36 Outcomes Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36.

Probability models allow us to find the probability of any collection of outcomes. Definition: An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. If A is any event, we write its probability as P(A). In the dice-rolling example, suppose we define event A as “sum is 5.” There are 4 outcomes that result in a sum of 5. Since each outcome has probability 1/36, P(A) = 4/36. Suppose event B is defined as “sum is not 5.” What is P(B)? P(B) = 1 – 4/36 = 32/36

Tree diagram A easy way to organize outcomes Helps to make sure that we don’t overlook any outcomes Ex: Create a tree diagram for flipping a coin and rolling a die Shown on next slide

Probability Rule #1 Probability Rule #2 Any probability is a number between 0 and 1 0 ≤ P(A) ≤ 1, A is an event All possible outcomes together must have probability 1 If S is the sample space, P(S) = 1 Probability Rule #2

P(S) = 1 The sum of the probabilities of all possible outcomes within the sample space must equal one Using the sample space for the sum of two die Sum of Die Probability of event 2 3 4 5 6 7 8 9 10 11 12

Probability Rule #3 Probability Rule #4 The probability that an event does not occur is 1 minus the probability that the event does occur (complement) P(Ac) = 1 – P(A) If two events have no outcomes in common (disjoint), the probability that one or the other occurs is the sum of their individual probabilities P(A or B) = P(A) + P(B) (addition rule) Probability Rule #4

Complement Rule P(Ac) = 1- P(A) The probability an event does not occur is 1 minus the probability it does occur Ex: If the probability of you getting a 5 on the AP exam is 12%, what is the probability that you do not get a 5?

Set Notation {AB} “A union B” The set of all outcomes that are either in A or B {AB} “A intersect B” The set of all outcomes that are in both A and B  “empty set” If two sets are disjoint (mutually exclusive) then AB =  Venn Diagrams

4. Disjoint events or Mutually exclusive events Events that cannot occur together are called mutually exclusive or disjoint events Have no outcomes in common so can never occur simultaneously EX. Getting an A and C in statistics the same 6 weeks. EX: Rolling a sum of 7 or 11 EX: Marital status of women ( divorced, single, married, widowed) P(A or B)= P(A) + P(B)

Example of disjoint events If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probability EX: Suppose the probability that a randomly selected students is a sophomore is.20 and the probability that he or she is a junior is .3. What is the probability that the student is either a sophomore or a junior? Could these outcomes occur at the same time? EX: What is the probability of rolling a 7 or 11?