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**Chapter 9**Introducing Probability**The Idea of Probability**Example: A random sample of n = 100 children has 8 individuals with asthma. What is the probability a child has asthma? • Probabilityhelps us deal with “chance” • Definition: the probability of an event is its expected proportion in an infinite series of repetitions ANS: We do not know. Although 8% is a reasonable “guesstimate,” the true probability is not known because our sample was not infinitely large**How Probability BehavesCoin Toss Example**Chancebehavior is unpredictable in the short run, but is predictable in the long run. The proportion of heads approaches 0.5 with many, many tosses.**Probability Models**• Probability models consist of these two parts: • Sample Space (S) = the set of all possible outcomes of a random process • Probabilities (Pr) for each possible outcome in the sample space Example of a probability model “Toss a fair coin once” S = {Heads, Tails} all possible outcomes Pr(heads) = 0.5 and Pr(tails) = 0.5probabilities for each outcome**0 ≤ Pr(A) ≤ 1**Pr(S) = 1 Addition Rule for Disjoint Events Law of Complements 4 Basic Rules of Probability (Summary) Also on bottom of page 1 of Formula Sheet**Rule 1 (Range of Possible Probabilities)**Let A ≡ event A Pr(A) ≡ probability of event A Rule 1 says “0 ≤ Pr(A) ≤ 1” Probabilities are always between 0 & 1 Pr(A) = 0 means A never occurs Pr(A) = 1 means A always occurs Pr(A) = .25 means A occurs 25% of the time Pr(A) = 1.25 Impossible! Must be something wrong Pr(A) = some negative number Impossible! Must be something wrong**Rule 2 (Sample Space Rule)**Let S ≡ the Sample Space Pr(S) = 1 All probabilities in the sample space must sum to 1 exactly. Example: “toss a fair coin” S = {heads or tails} Pr(heads) + Pr(tails) = 0.5 + 0.5 = 1.0**Rule 3 (Addition Rule, Disjoint Events)**Events A and B are disjoint if they can never occur together. When events are disjoint: Pr(A or B) = Pr(A) + Pr(B) Age of mother at first birth Let A ≡ first birth at age < 20: Pr(A) = 25% Let B ≡ first birth at age 20 to 24: Pr(B) = 33% Let C ≡ age at first birth ≥25 Pr(C) = 42% Probability age at first birth ≥ 20 = Pr(B or C) = Pr(B) + Pr(C) = 33% + 42% = 75%**Rule 4 (Rule of Complements)**Let Ā≡ A does NOT occur This is called the complement of event A Pr(Ā) = 1 – Pr(A) Example: If A ≡ “survived” then Ā ≡ “did not survive” If Pr(A) = 0.9 then Pr(Ā) = 1 – Pr(A) = 1 – 0.9 = 0.1**Probability Mass Functions pmfs**Probability mass functions are made up of a list of separated outcomes. For discrete random variables. Example of a pmf: A couple wants three children. Let X ≡ the number of girls they will have Here is the pmf that suits this situation:**To assign probabilities for continuous random variables **we density curve Properties of a density curve Always on or above horizontal axis Has total area under curve (AUC) of exactly 1 AUC in any range = probability of a value in that range Probability Density Functions pdfs (“Density Curves”) Probability density functions form a continuum of possible outcomes. For continuous random variables.**Note**The curve is always on or above horizontal axis and has AUC = height × base = 1 × 1 = 1 Probability = AUC in the range. Examples follow. Pr(X < .5) = height × base = 1 × .5 = .5 Pr(X > 0.8) = height × base = 1 × .2 = .2 Pr (X < .5 or X > 0.8) = .5 + .2 = .7 Example of a pdf This random spinner has this pdf density“curve”**pdf Density Curves**• Density curves come in many shapes • Prior slide showed a “uniform” shape • Below are “Normal” and “skewed right” shapes • Measures of center apply to density curves • µ (expected value or “mean”) is the center balancing point • Median splits the AUC in half**From Histogram to Density Curve**Histograms show distribution in chunks The smooth curve drawn over the histogram represents a Normal density curve for the distribution 8/14/2014 15**Area Under the Curve (AUC)**Area in Bars = proportion in that range 30% of students had scores ≤ 6 30% of students had scores ≤ 6 Shaded area = 30% of total area of the histogram 30% 70% 8/14/2014 16**Area Under the Curve (AUC)**Area Under Curve = proportion in that range! 30% of students had scores ≤ 6 30% of area under the curve (AUC) is shaded 30% 70% 8/14/2014 17**Summary of Selected Points**• To date we have studied descriptive statistics. From here forward we study inferential statistics {2} • Probability is the study chance; chance is unpredictable in the short run but is predictable in the long run {3 - 4}; take the rules of probability to heart{5 - 10} • Discrete random variables are described with probability mass function • Continuous random variables are described with density curves with the area under the curve (AUC) corresponding to probabilities