The Study of Randomness Probability The Study of Randomness

The language of probability Random in statistics does not mean “haphazard”. Random is a description of a kind of order that emerges only in the long run even though individual outcomes are uncertain. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Takes in the idea of probability. Hits 6.1 pg 331

Probability Models The sample space of a random event is the set of all possible outcomes. What is the sample space for rolling a six-sided die? S = {1, 2, 3, 4, 5, 6} What is the sample space for flipping a coin and then choosing a vowel at random? Section 6.2 pg 336 box of vocabulary. S = samples space s = standard deviation. Remind them about a e I o u and do we use y?

Tree diagram S={Ha, He, Hi, Ho, Hu, Ta, Te, Ti, To, Tu} a e H i o u a This is where we can bring up the “multiplication rule” T S={Ha, He, Hi, Ho, Hu, Ta, Te, Ti, To, Tu}

What is the sample space for answering one true/false question? S = {T, F} What is the sample space for answering two true/false questions? S = {TT, TF, FT, FF} What is the sample space for three?

Tree diagram S = {TTT, TTF, TFT, FTT, FFT, FTF, TFF, FFF} True True False True True False False True True False False True False False S = {TTT, TTF, TFT, FTT, FFT, FTF, TFF, FFF}

Intuitive Probability An event is an outcome or set of outcomes of a random phenomenon. An event is a subset of the sample space. For probability to be a mathematical model, we must assign proportions for all events and groups of events. Still in section 6.2 on page 336 of book.

Basic Probability Rules The probability P(A) of any event A satisfies 0 < P(A) < 1. Any probability is a number between 0 and 1, inclusive. If S is the sample space in a probability model, then P(S) = 1. All possible outcomes together must have probability of 1. Probability rule worksheet to fill in. Rule 1 is the probability is a number between 0 and 1 and Rule 2 all outcomes together must have prob of 1

Complement Rule The complement of any event A is the event that A does not occur, written as Ac. The complement rule states that P(Ac) = 1 – P(A) The probability that an event does not occur is 1 minus the probability that the event does occur. Rule 3 on Notes Rule Worksheet On page 343 in book The little “c” means complement

Venn diagram: complement S Ac Make them draw and show it on their rule sheet

General Addition Rule for Unions of Two Events For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) P(AB) = P(A) + P(B) – P(AB) The simultaneous occurrence of two events is called a joint event. The union of any collections of event that at least one of the collection occurs. Rule 4 (not in book until later) section 6.3 page 360 and the General rule addition is on page 362 Introduction of or and the and “union” How to know And  looks like an N which is in the word “and” The “U” rocks back and forth so it means “or”

Venn diagram: {A and B} S A B This explains the formula of why you subtract the P(a and b) The S – “sample space” in the top corner is not taking up 100% of sample space there is a neither a or b in the square of the venn diagram Get students to recognize the space outside the A and B

Venn diagram: disjoint events (Mutually Exclusive) B A This will lead us to the Addition Rule. There is still 2 events with no AND.

Addition Rule Two events A and B are disjoint (also called Mutually Exclusive) if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint, P(A or B) = P(A) + P(B) If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Mutually Exclusive is a big deal (book doesn’t mention much about it) Section 6.2 pg 343 figure 6.4 Under rule 4 write this under “special case” for the general rule

General Multiplication Rule The joint probability that both of two events A and B happen together can be found by P(A and B) = P(A) P(B|A) P(B|A) is the conditional probability that B occurs given the information that A occurs. General addition and multiplication are the only 2 on the formula chart. Introduce B occurs given that A

Definition of Conditional Probability When P(A)>0, the conditional probability of B given A is OR P(AB) = P(AB) P(B) This is a big deal when we discuss independence The denominator can not equal 0

Multiplication Rule If one event does not affect the probability of another event, the probability that both events occurs is the product of their individual probabilities. Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) Definition of Independent event “special cases” Only occurs when not “given that”

Question #3 P(A or B) = P(A) + P(B) – P(A and B) Suppose that 60% of all customers of a large insurance agency have automobile policies with the agency, 40% have homeowner’s policies, and 25% have both types of policies. If a customer is randomly selected, what is the probability that he or she has at least one of these two types of policies with the agency? (Hint: Venn diagram) P(A or B) = P(A) + P(B) – P(A and B) The Venn Diagram needs to be drawn to help illustrate the Addition Rule. P(auto or home) = .60 + .40 - .25 = .75

Question #6 Drawing two aces with replacement. Drawing three face cards with replacement. Many students have never played a card game with a real deck of cards, only electronic games. Be sure to explain the different suits and types of cards in a deck. Remind the students that an Ace is neither a number card or face card.

Multiplication Rule Practice Draw 5 reds cards without replacement. Draw two even numbered cards without replacement.

Multiplication Rule Practice Draw three odd numbered red cards with replacement.

Back to Flipchart

Question #7 What is the probability of a GFI switch from a selected spa is from company 1? Use this question to teach conditional probability. What is the probability of a GFI switch from a selected spa is defective?

Question #7 What is the probability of a GFI switch from a selected spa is defective and from company 1? What is the probability of a GFI switch from a selected spa is from company 1 given that it is defective?

Question #7 P(A and B) = P(A) P(B|A)

Remember Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) The joint probability that both of two events A and B happen together can be found by P(A and B) = P(A) P(B|A) How can we use the formulas to test for independence?

Independence and Mutually Exclusivity Independence means knowing something about one tells you nothing about the other. Mutually exclusive events cannot happen at the same time. Are independent events mutually exclusive?

Independent Events Two events A and B that both have positive probability are independent if P(B|A) = P(B) Back to flipchart

Yes No Total Male 21 14 35 Female 39 26 65 60 40 100 13. Jack and Jill have finished conducting taste tests with 100 adults from their neighborhood. They found that 60 of them correctly identified the tap water. The data is displayed below. Yes No Total Male 21 14 35 Female 39 26 65 60 40 100 Use this question to teach the proper procedure for determining if 2 events are independent. Is the event that a participant is male and the event that he correctly identified tap water independent?

In order for a participant being male and the event that he correctly identified tap water to be independent, we know that Yes No Total Male 21 14 35 Female 39 26 65 60 40 100 P(male|yes) = P(male) or P(yes|male) = P(yes)

In order for a participant being male and the event that he correctly identified tap water to be independent, we know that P(yes|male) = P(yes) Yes No Total Male 21 14 35 Female 39 26 65 60 40 100

In order for a participant being male and the event that he correctly identified tap water to be independent, we know that P(male|yes) = P(male) Yes No Total Male 21 14 35 Female 39 26 65 60 40 100

Question #16 What is the probability of an individual having tuberculosis given the DNA test is negative?

Conditional Probability with Tree Diagrams 17. Dr. Carey has two bottles of sample pills on his desk for the treatment of arthritic pain. He often grabs a bottle without looking and takes the medicine. Since the first bottle is closer to him, the chances of grabbing it are 0.60. He knows the medicine from this bottle relieves the pain 70% of the time while the medicine in the second bottle relieves the pain 90% of the time. What is the probability that Dr. Carey grabbed the first bottle given his pain was not relieved? This problem is the first one to use a tree diagram for solving conditional probability. You can change this problem into a table to find the conditional probability, like the table in questions #12 and 13.

relieved not .7 1st .6 .3 relieved not .9 .4 2nd .1