UNIT 5: PROBABILITYSample Space

Rock, Paper, Scissors

Tournament

History

• As old as civilization.• Egyptians used a small mammal bone as a 4 sided die (3500 BC)

• Games of chance were common in Greek & Roman times

• Formal study – began with Blaise Pascal & Pierre de Fermat due to gambling problems.

Activity (1 in 6)

1. Roll your die seven times to imitate the process of the seven friends buying their soda. How many of them won a prize.

2. Repeat step 1 four more times. In the 5 repetitions, how many times did three or more of the group win a prize?

3. Combine results with your classmates. What percent of the time did the friends come away with three or more prizes, just by chance?

4. Based on your answer in step 3, does it seem plausible that the company is telling the truth, but that the seven friends just got lucky?

Probability

•It’s the long-run relative frequency.

Empirical Probability

• This is the relative frequency that is obtained by actually doing the experiment.

Activity: Coin Toss - Graph

Law of Large Numbers

•With more repetitions, the empirical probability gets closer to the theoretical probability.

•Theoretical Probability is based on what I should get (in theory)….based on the sample space.

Probability• It is the outcome of a chance process that is a number

between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

Life Insurance:We can’t predict whether a particular person will die in the next year. But, the National Center for Health Statistics says that the proportion of men aged 20 to 24 years who die in any one year is 0.0015.

*What percent of men (20 to 24) will die?

*The probability of a woman (20 to 24) will die is one-third that of a man. What percent of women (20 to 24) will die?

*Would you be surprised to learn that they charge a man (20 to 24) three times more than a woman (20 to 24)?

According to the “Book of Odds,” the probability that a randomly selected U.S. Adult usually eats breakfast is 0.61.• What does the probability 0.61 mean in this setting?

• Does this mean that if 100 U.S. adults are chosen at random, exactly 61 of them usually eat breakfast?

Match the following probabilities with each statement…

0 0.01 0.3 0.6 0.99 1

•This outcome is impossible…it can never occur.

•This outcome is certain. It will occur on every trial.

•This outcome is very unlikely, but it will occur once in a while in a long sequence of trials.

•This outcome will occur more often than not.

Which of the following outcomes is more probable?

HTHTTH TTTHHH

Myth of Short-run Regularity

• Probability is predictable in the long run…not the short run.

Myth of the “Law of Averages”

• Things do even out…in the long run.

• If I tossed a coin and got TTTTTT, which of the following is more likely on the next toss?

• Getting Heads

• Getting Tails

Simulation

• It is an imitation of chance behavior, based on a model that accurately reflects the situation.

•This is what we did in the 1 in 6 game earlier

At a local high school, 95 students have permission to park on campus. Each month, the student council holds a “golden ticket parking lottery” at a school assembly. The two lucky winners are given reserved parking spots next to the school’s main entrance. Last month, the winning tickets were drawn by a student council member from the AP Stats class. When both golden tickets went to members of that same class, some people thought the lottery had been rigged. There are 28 students in the AP Stats class, all of whom are eligible to park on campus. Design and carry out a simulation to decide whether it’s plausible that the lottery was carried out fairly.

Let’s start at line 139 of the Random number table. Let the students who can park be represented as 01 to 95 and the students in the AP stats is 01 to 28. We will simulate drawing two tickets. We will use 18 trials.

55588 99404 70708 41098 43563

56934 48394 51719

In an attempt to increase sales, a breakfast cereal company decides to offer a NASCAR promotion. Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale, Earnhardt, Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson. The company says that each of the 5 cards is equally kiely to appear in any box of cereal. A NASCAR fan decides to keep buying boxes of the cereal until she has all 5 drivers’ cards. She is surprised when it takes her 23 boxes to get the full set of cards. Should she be surprised? Design and carry out a simulation to help answer this question.

Let 1-5 represent finding each of the five NASCAR drivers. Since we want a full deck, then we will go until we get all 5 cards. We will keep up with the number of boxes that we had to check in order to get all 5. Let’s start with line116.

14459 26056 31424 80371

65103 62253 50490 61181

38167 98532 62183 70632

23417 26185 41448 75532

Activity: Is it Fair?

Homework

• Page 293 (1, 3, 7, 9, 13, 17, 19, 23, 25)