MTH 245: Mathematics for Management, Life, and Social Sciences

Transcription

1 1/13 MTH 245: Mathematics for Management, Life, and Social Sciences Section 6.2

2 Section 6.1: Assignment of Probabilities. 2/13 Create the probability distribution for the experiment of rolling a die and recording the number of dots on the up side. Outcome Probability

3 Section 6.1: Assignment of Probabilities. 3/13 Probabilities can be assigned by observing a (large) number of trials and using the frequency of outcomes to estimate probability. Let A be any event defined on a sample space S, the symbol A will denote the probability of A. Axiom 1. Let A be any event defined over S. Then P(A) 0. Axiom 2. P(S) = 1 Axiom 3. Let A and B be any two mutually exclusive events defined over S. Then, P(A B) = P(A) + P(B)

4 Section 6.1: Assignment of Probabilities. 4/13 In terms of simple events We describe the latter axioms in terms of simple events: Let S = {s 1,...,s n } be a sample space. We assign to each outcome s i in the sample space a probability P(s i ). The probability of event E,P(E), is the sum of the probabilities of all simple events making up E. Ex. If E = {s 3,s 4,s 6 } then P(E) = P(s 3 ) + P(s 4 ) + P(s 6 ). Since 0 P(E) 1 for any event E defined in S, then, 0 P(s i ) 1 for all i. The sum of probabilities of all simple events adds up to 1. That is, P(S) = P(s 1 ) + P(s 2 ) + + P(s n ) = 1.

5 Section 6.1: Assignment of Probabilities. 5/13 Example 1 Consider the sample space S = {s 1,s 2,s 3,s 4,s 5,s 6 } with assigned probabilities P(s 1 ) = 0.05, P(s 2 ) = 0.25, P(s 3 ) = 0.05, P(s 4 ) = 0.01 P(s 5 ) = 0.63, P(s 6 ) = Let E = {s 1,s 2 } and F = {s 3,s 5,s 6 }. Determine P(E), P(F) and P(E F). 1. P(E) = P(s 1 ) + P(s 2 ) = 2. P(F) = 3. P(E F) =

6 Section 6.1: Assignment of Probabilities. 6/13 Example 2 Which of the following probability assignments is valid? 1. S = {high,medium,low} P(high) = 0.3, P(medium) =.4, P(low) =.3 2. S = {blue,purple,green,pink} P(blue) =.4, P(purple) =.5 P(green) =.6, P(pink) =.3 3. S = {rain,sleet,snow} P(rain) = 1.2, P(sleet) = 2.3, P(snow) = S = {hot,mild,cool,cold} P(hot) =.35, P(mild) =.38 P(cool) =.31, P(cold) =.33

7 Section 6.1: Assignment of Probabilities. 7/13 Inclusion-Exclusion Principle Inclusion Exclusion Principle Let E and F any events. Then P(E F) = P(E) + P(F) P(E F) Note: As we mentioned before, note that P(E F) = P(E) + P(F) is F and E are mutually exclusive.

8 Section 6.1: Assignment of Probabilities. 8/13 Example 3 Let A and B be two events defined on a sample space S such that P(A) = 0.3, P(B) = 0.5 and P(A B) = 0.7. Find P(A B).

9 Section 6.1: Assignment of Probabilities. 9/13 Example 4 In a newly released martial arts film, the actress playing the lead role has a stunt double who handles all of the physically dangerous action scenes. According to the script, the actress appears in 40% of the film scenes, her double appears in 30%, and the two of them are together 5% of the time. What is the probability that in a given scene neither the lead actress nor the double appears? Note: It would be useful to show that for any event A, P(A ) = 1 P(A). We start by noticing that A and A are mutually exclusive...

10 Section 6.1: Assignment of Probabilities. 10/13 Odds vs. Probability Odds vs. Probability: Probability expresses the likelihood of an event occurring as a number from 0 to 1. Odds express this in another way: a pair of integers that contrast the chances in favor of the event to the chances against the event occurring. (Note that these are NOT gambling odds, as those are odds against, and are often calculated in a way that assumes a certain payoff to the house)

11 Section 6.1: Assignment of Probabilities. 11/13 Converting between Odds and Probabilities. If the odds in favor of the event E occurring are a to b then P(E) = a a + b. If P(E) = p, then the odds in favor of E are found by reducing the p fraction 1 p to the form a where both a and b are integers not having b common divisor. Then the odds in favor of E occurring are a to b.

12 Section 6.1: Assignment of Probabilities. 12/13 Car Ownership: Exercise 46 Example 5 The odds of an adult in the United States owning a passenger are 39 to 12. What is the probability that and adult in the United States owns a passenger car?

13 Section 6.1: Assignment of Probabilities. 13/13 Example 6 The probability that there will be a major earthquake in the San Francisco area during the next 30 years is.63. What are the corresponding odds?