Probability

1. Probability The Study of Chance!

2. When we think about probability, most of us turn our thoughts to games of chance

3. Probability is also the underlying foundation for the methods we use in inferential statistics. Let’s learn some basics

4. PROBABILITY BASICS Any action with unpredictable outcomes EXPERIMENT: SAMPLE SPACE: The set of all possible outcomes “S” EVENT: A specific outcome P (A): The probability of outcome A occurring Complement: The complement of any event “A”, consists of all outcomes in which “A” does NOT occur.

5. PROBABILITY BASICS • A probability is a number between 0 and 1 • For any event A, 0 ≤ P(A) ≤ 1 • The probability of the set of all possible outcomes (sample space) must be 1. • P(S)=∑P(A) = 1 • This is known as • “The something’s gotta happen rule” • The probability of an impossible event is 0 • Most events have probabilities somewhere between! • The probability of an event that is certain is 1

6. Approaches toProbability • Relative Frequency Approximation: (Experimental Probability) • Conduct the experiment a large number of times and record the results Then:

7. ForExample ??What’s the probability that I roll a 4 with a standard die (A) ?? To answer this using relative frequency approach (experimental probability) I rolled a die 200 times. Outcomes

8. 2nd Approach toProbability • Classical Approach to Probability (Theoretical Approach) **Requires Equally-likely Outcomes** • Assume that a procedure has “n” different simple events and each of those events has an equal chance of happening Then:

9. ForExample ??What’s the probability that I roll a 4 with a standard die (A) ?? Let’s answer this using the relative frequency approach (theoretical probability) • A standard die has one “4” • A standard die has 6 possibilities

10. So, let’s summarize Experimental P(4) = .145 Theoretical P(4) = .166 Why did we get two different answers? Probability is really about what would happen “in the long run”. The “Law of Large Numbers” tells us that experimental probability will approach the actual (theoretical) probability in a LARGE number of trials. If we increased the number of rolls our probability would get closer and closer to .166

11. 3rd Approach toProbability • Subjective Probability • Probabilities are found by simply guessing or estimating its value based on prior knowledge of the relevant circumstances • Examples of subjective probability include forecasting the weather

12. Complementary Events • The complement of an event is when the event does NOT happen. Remember that one of our basics of probability is the “something’s gotta happen rule” which says that the sum of all possibilities equals 1. So, the probability of a complement (A’) can be found with 1 – the probability that the event happened. P(A’) = 1 – P(A)

13. For Example For a single card drawn from a standard deck of cards, what is the probability that it is NOT a diamond? We can use what we know about complementary events to answer this question. Facts about cards: 13 diamonds 13 hearts 13 spades 13 clubs Call “diamond” event A P(A’) = 1 – P(A) P(A’) = 1- (13/52)= .75

14. Multiple Events • Consider the event: • Tossing a coin followed by rolling a standard die • We want to calculate the probability of rolling a 5. • To find this probability, consider the following model.

15. We will use a tree diagram to represent the stages in the event. In order to keep the “tree” manageable, we will define each stage as simply as we can. i.e.—”success” or “not a success” The second stage was to roll a die. Although there are 6 possible outcomes, we are only interested in the outcome “roll a 5”. We can simplify our tree by designating only “success”—5, and not a success—5c Remember: Our question is P(rolling a 5) = ? The first stage has two possible outcomes 5 P(heads & 5) = Heads 5c Notice this happens in two places in our tree. Toss a coin P(tails & 5) = 5 Tails To find the probabilities for these two “branches”, we multiply the individual probabilities across and then add the two branches together 5c

16. Calculating Probability • This means that the probability of rolling a “5” is calculated by: • = (1/2)(1/6) + (1/2)(1/6) • = (1/12) + (1/12) • = (2/12)

17. Multiple Events and a couple of shortcuts • Let’s consider the following situation • Roll a standard die 3 times • Now find the following probabilities • What is the probability that we get one “4” in 3 rolls? • What is the probability that not all rolls are “4’s” • What is the probability that we get at least one “4”?

18. Draw a Picture Find the branches where this happens. Find the probabilities for each branch 4 Find the probability that we get one “4” in three rolls. 4 4c 4 4 4c 4c (1/6)(5/6)(5/6)=.1157 Roll a Die 4 4 4c (5/6)(1/6)(5/6)=.1157 4c 4 (5/6)(5/6)(1/6)=.1157 .3471 4c Then find the sum of the probabilities of these branches 4c 2nd Roll 3rd Roll

19. Draw a Picture 4 1-[(1/6)(1/6)(1/6)]=.9954 Find the probability that not ALL are “4’s” 4 4c Find the branches where this happens 4 4 4c 4c Find the probabilities for each branch OR notice that all branches except the branch where all are 4’s are checked….. So….we could find the P(not all) by using complements 1- P(all) Roll a Die 4 4 4c 4c 4 4c 4c 2nd Roll 3rd Roll

20. Draw a Picture Find the branches where this happens 4 Find the probability that at least one is a “4” 4 Find the probabilities for each branch OR notice that all branches except the branch where none are 4’s are checked….. So….we could find the P(at least one) by using complements 1- P(none) 4c 4 4 4c 4c Roll a Die 4 4 4c 4c 4 4c 4c 1-[(5/6)(5/6)(5/6)]=.4213 2nd Roll 3rd Roll

21. Additional Resources • The Practice of Statistics—YMM • Pg 310 -322 • The Practice of Statistics—YMS • Pg 328-355 • Against All Odds—Video #15 • http://www.learner.org/resources/series65.html