Defining Probabilities: Random Variables Examples: Out of 100 heart catheterization procedures performed at a local hospital each year, the probability that more than five of them will result in complications is __________ Drywall anchors are sold in packs of 50 at the local hardware store. The probability that no more than 3 will be defective is In general, ___________ P(X > x) P(Y < 3)

Discrete Random Variables Example: Look back at problem 3, page 46. Assume someone spends $75 to buy 3 envelopes. The sample space describing the presence of $10 bills (H) vs bills that are not $10 (N) is: _____________________________ The random variable associated with this situation, X, reflects the outcome of the choice and can take on the values: Note: if the number of possible solutions is countable, the variable is discrete S = {NNN, NNH, NHN, HNN, NHH, HNH, HHN, HHH} X = {0, 1, 2, 3}

Discrete Probability Distributions The probability that there are no $10 in the group is P(X = 0) = ___________________ (recall results from last time) The probability distribution associated with the number of $10 bills is given by: P(X=0) = P(not in the 1st envelope ∩ not in the 2nd ∩ not in the 3rd) = (1-275/500)3 = (0.45)3 = 0.09112 P(0) =(1-0.55)^3 = 0.091125 P(1) =3*((0.55)*(1-0.55)^2) = 0.334125 P(2) =3*(0.55^2*(1-0.55)) = 0.408375 P(3) = 0.55^3 = 0.166375 (students fill in the table) x 1 2 3 P(X = x)

Another Example Example 3.3, pg 66 P(X = 0) = _____________________ P(X = 0) = P(0 defectives and 2 nondefective) = (all ways to get 0 out of 3 defectives) ∩ (all ways to get 2 out of 5 nondefective) (all ways to choose 2 out of 8 computers) (all ways to choose 2 out of 8 computers)

Discrete Probability Distributions The discrete probability distribution function (pdf) f(x) = P(X = x) ≥ 0 Σx f(x) = 1 The cumulative distribution, F(x) F(x) = P(X ≤ x) = Σt ≤ x f(t)

Probability Distributions From our example, the probability that no more than 2 of the envelopes contain $10 bills is P(X ≤ 2) = F(2) = _________________ The probability that no fewer than 2 envelopes contain $10 bills is P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - F(1) = ________________ F(2) = f(0) + f(1) + f(2) = .833625  (OR 1 - f(3)) 1 – F(1) = 1 – (f(0) + f(1)) = 1 - .425 = .575  (OR f(2) + f(3))

Another View The probability histogram

Your Turn … The output from of the same type of circuit board from two assembly lines is mixed into one storage tray. In a tray of 10 circuit boards, 6 are from line A and 4 from line B. If the inspector chooses 2 boards from the tray, show the probability distribution function associated with the selected boards being from line A. x P(x) 1 2 Hint: draw the tree

Continuous Probability Distributions Examples: The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is __________ The probability that a given part will fail before 1000 hours of use is In general, __________ Probability density function f(x)

Homework due dates Friday, 9/3 pg.54-56 Wednesday, 9/8 pg. 72-74 (both sets) (Check the web site on Thursday, 9/2, for an updated schedule.)