Math Tech IIII, Jan 21

Transcription

1 Math Tech IIII, Jan 21 Probability III The Complement of an Event, Theoretical and Experimental Probability Book Sections: 3.1 Essential Questions: How can I compute the probability of any event? What is the compliment of and an event and what is its probability? What is the difference between probability theory and reality and how does it affect probability? Standards: DA-1.5, DA-5.5, DA-5.9, DA-5.7, S.CP.1

2 Complementary Events The compliment of an event A is everything happening except A The compliment of A is called not A and is abbreviated with A (called as A bar)

3 Examples The compliment of heads on a coin flip would be not heads - which would be tails The compliment of a 5 on a dice roll (not 5) - would be 1, 2, 3, 4, or 6 The compliment of a rainy day forecast for today - would be no rain today

4 Complementary Events An event will either happen or it will not. All possible outcomes that are not an event add up to be the compliment of that event. The sum of the probability of an event and the event s compliment always add up to 1 or P(event) + P(not the event) = 1, where P(not the event) is the probability of the event s compliment.

5 In Other Words The probability of a complimentary event is: P(event) = 1 P(event) or P(A) = 1 P(A) Example: What is the probability of not getting a 3 when rolling a die? 1 P(3) =

6 Examples

7 The Mathematical Definition of Probability One More Time P(event) = Number of favorable outcomes Total number of outcomes In words: The probability of an event is the ratio of favorable outcomes to the number of possible outcomes. That number will always be between 0 and 1.

8 The Basis of Probability Theory To date, every probabilistic model we have considered (dice, coins, spinners, cards, ect) has been based on what should happen if we had true random events and how we would compute probabilities. Probabilities that are based on known characteristics or facts are called theoretical probabilities, and that is what we have studied so far in this unit.

9 Types of Probability Theoretical Probability Probability based on a mathematical model, or what should happen in random events. Experimental Probability - Probability based on repetitions of an actual experiment, computed from the results of a lot of observations, or what actually happens in random events.

10 Computing Experimental Probability Experimental probability is computed from manymany actual observations P(event) = Number of favorable outcomes observed Total number of Trials

11 Theory Can Only Cover so Many Bases If we have complete knowledge of the set of possible outcomes of an event, then theoretical probability is a good predictor of what should happen. If we don t know how an experiment will behave or the outcome of event will occur, there is no way of predicting results based on theories, so we have to gain that knowledge by conducting the experiment many times.

12 Experimental Probability What it Means Experimental probability is a probability based on conducting an experiment over and over and can vary as the experiment is continually repeated.

13 An Example Over the last eight years, Farmer Jones has determined that only five out of six corn seeds planted on his 3000 acre farm produce corn. Is this theoretical or experimental probability? Experimental because it is based on what happened in the past. If Farmer Jones wants 10,000 corn-bearing plants, how many should he plant?

14 An Example Over the last eight years, Farmer Jones has determined that only five out of six corn seeds planted on his 3000 acre farm produce corn. Is this theoretical or experimental probability? Experimental because it is based on what happened in the past. If Farmer Jones wants 10,000 corn-bearing plants, how many should he plant? 5 of 6 seeds should produce corn ,000 x Set up the proportion 10,000 out of x seeds should produce corn or 5 x = 6 10,000, solving, x = 12,000 seeds

15 Example Two Quality control at a fan belt factory randomly selects 200 production belts a day and tests their quality. Over a ten day period, they have found a total of 15 defective fan belts. Experimental because it is based on sampling and testing (experimenting). What is the experimental probability that any given fan belt is defective from this factory?

16 Example Two, Answer What is the experimental probability that any given fan belt is defective from this factory? 200 belts/day 10 days = 2000 belts tested 15 bad out of 2000 = 15/2000 = 3/400 =.0075 The P(Bad Belt) =.0075 or.75% If the factory produces 25,000 belts every day, how many are defective? 25,000 x.0075 = or 188 bad belts Do you think the.75% probability is a constant?

17 Theoretical and Experimental Side-by Side 700 Trials of this spinner yield the following results: OC = Outcome OC Obs P(#) Theoretical P /140 (.136) /350 (.146) /7 (.143) /175 (.131) /70 (.129) /350 (.163) /700 (.153)

18 The Theory vs. Application Celebrity Death Match Does Theoretical Probability Match Experimental Probability? We shall see next time.

19 Class work: CW 1/21, All of it Homework: None