MATH 112 Section 7.3: Understanding Chance Prof. Jonathan Duncan Walla Walla University Autumn Quarter, 2007
Outline 1 Introduction to Probability 2 Theoretical vs. Experimental Probability 3 Advanced Probability Topics 4 Conclusion
What is Probability? To introduce us to probability, we start with an example. Suppose that 5000 people take a standardized test and that 5 of them score below 38 on the test. What is the probability that if you pick one of the 5000 test takers at random they scored less than 38? Parts of a Probability Problem The previous probability problem has the following parts: Outcome - The choice of a particular exam taker Sample Space - The 5000 people from which we choose Event - Choosing any of the 5 people below 38
Equally Likely Outcomes In the first example, each outcome is equally likely. That is, we are just as likely to choose person A as person B. Probability and Equally Likely Outcomes If E is an event in a sample space S and each outcome in S is equally likely, then P(E) = outcomes in E outcomes in S To finish our example, the probability of picking a person with a score of less than 38 is: 5 5000 = 1 1000 = 0.001
Probability and Events Let s continue our discussion of probability with another example. Suppose that you roll a die in which the numbers 1 and 5 are weighted so as to come up 1 3 of the time each and all other numbers come up 1 12 of the time. Find the probability that in a single roll the number will be odd. Addition Rule If an event E consists of n distinct outcomes O 1, O 2,..., O n then P(E) = P(O 1 ) + P(O 2 ) + P(O n ). Although the outcomes for our die are not equally likely, we can compute P(Odd) = P(1) + P(3) + P(5) = 1 3 + 1 12 + 1 3 = 9 12 = 3 4.
Some Probability Rules Probabilities have certain rules which they must follow. Probability Rules Let S be a sample space with outcomes O 1, O 2,..., O n. Let E be an event in the sample space. Then, the following rules hold: P(S) = P(O 1 ) + P(O 2 ) + + P(O n ) = 1 0 P(E) 1 If P(E) = 1 then E is certain to happen If P(E) = 0 then E is certain not to happen You will all be given a coin and asked to flip it 4 times. Before you flip your coins, what do you think the probability is that you will get 2 heads and 2 tails?
Experimental Probability We will now work as a class to determine the experimental probability of flipping two heads and two tails in four coin flips. After each student flips his or her coins, record the number of heads each student flips. Finally, determine the percent of students who flipped 2 heads. Experimental Probability To determine an experimental probability, an experiment is repeated multiple times. The probability of an event is the number of times the event took place divide by the number of times the experiment was repeated.
Theoretical Probability Many times it is not possible to compute a probability experimentally. Instead, we use theoretical probability. Using a tree diagram (as seen in the multiplication section) determine the probability of flipping two heads in four flips of a fair coin. Experimental vs. Theoretical Probability Give at least two advantages of experimental probability and two advantages of theoretical probability.
More Theoretical Probabilities Using the same tree as we used in the previous example, determine the probability of getting at least one heads in four flips. Mutually Exclusive Events If two events can not happen at the same time, they are called mutually exclusive. The probability of the union of these events is the sum of the probability of the events. The probability of at least one heads in four flips can be found by: P(1 head) + P(2 head) + P(3 head) + P(4 head)
Another Approach While we managed to solve the example in the previous slide, it took a little bit of work. There is a short cut which could help us solve the problem more quickly. Complements If E is an event in a sample space S then the probability of E, the complement of E, is 1 P(E). To find the probability of at least one head in four flips, we will compute the probability of the complement: no heads in four flips. P(at least 1 heads) = 1 P(no heads) = 1 1 16 = 15 16
Binomial Probabilities If we examine probability questions in which a single task (called a trial) is repeated multiple times, such as the coin flipping we have just finished, we will find that there is a familiar pattern. You take a multiple choice quiz with 5 questions, each having 4 possible answers one of which is correct. If you have a 25% chance of guessing correctly, what is the probability you will answer 3 or more questions correctly? Binomial Probabilities The probability of r successes in n trials in which the probability of a success is p is given by ( ) n P(r) = p r (1 p) n r r
Expected Value We will finish with one last example of a probability question. This has to do with events in a sample space which have a certain value. A game consists of drawing 2 balls from an urn containing 10 balls: 7 black, 2 red, and one yellow. You are paid $5 for each yellow ball that you draw, $1 for each red ball, and nothing for each black ball. It costs you $1 to play. How much can you expect to make on each game? Caution Note that this is how much you would expect to make per game on average if you played many games. You will never actually win this amount in a single game.
Important Concepts Things to Remember from Section 7.3 1 Computing Probabilities with Equally Likely Events 2 Probability Rules 3 Theoretical vs. Experimental Probabilities 4 Binomial Probabilities 5 Expected Value