Prisoners of Probability

Fun Statistics ______ of Americans don’t know the people that live next door. less than 25% 25%-50% 50%-75% 75%-100% Answer: 72%

Fun Statistics About _____ Americans are injured by musical instruments each year. 2,000 5,000 8,000 10,000 Answer: 8,000

Fun Statistics ______ of pet owners sleep with their pets. less than 25% 25%-50% 50%-75% 75%-100% Answer: 63%

Fun Statistics Nearly ____ of all marriage proposals are made over the phone. less than 25% 25%-50% 50%-75% 75%-100% Answer: 6%

Fun Statistics _____ of men say that they wouldn’t mind being stupid as long as they had the perfect body. less than 25% 25%-50% 50%-75% 75%-100% Answer: 19%

Pascal’s Mathematical Theory of Probability Blaise Pascal played a role in the development of probability theory. As a child, he had a talent for math. Although his father forbade him from to pursue his interest, he changed his mind after finding him writing geometric proofs on a wall with a lump of coal. In 1642, Pascal, not yet 19, constructed a mechanical calculator capable of addition and subtraction. It became known as Pascal’s calculator or the Pascaline.

Pascal’s Mathematical Theory of Probability The calculator failed to be a huge commercial success because it so expensive. However, he continued to make improvements to his design and built dozens of more machines. Such machines raised the question: Can we create a mechanical process that thinks, and an algebraic algorithm that decides questions? To do this, one might study the mathematics of probability.

Multiplication Principle If I roll a dice, how many possible outcomes are there? What are they? 1,2,3,4,5,6 How about if I roll two dice?

Multiplication Principle You would have 36 possible outcomes. If I rolled three dice, there would be 216 possible outcomes. Multiplication Principle: If something can happen in M different ways, and something else can happen in N different ways, then there are M x N different ways these events can happen.

Using Math to Make Choices How do we make choices? Determine your goal. Determine its value. Arrange and examine the options available to reach it. Determine the likelihood of each option meeting your goal. Choose the option with the highest likelihood of meeting it. Use the outcome of this experience to adjust your future goals and the way in which you make future decisions. Can we make choices in a mathematical way? Do we without even realizing it?

The Prisoner’s Dilemma Two criminals are captured for a minor crime. They are suspected of having committed a major crime too. So, the police interrogate each of them separately, and make each the same offer: "If neither of you tells me you were involved in the big crime, you will each get 3 years in jail for the small crime. But consider the alternative: if one of you accuses the other, then the accuser will go free, and the other will get 10 years in prison. If you both accuse each other, however, you'll each get 5 years in prison. The police interrogate each prisoner separately, they cannot communicate. What would you do? Why?

The Prisoner’s Dilemma Which option has the better outcome? Why? Is it better if we make choices mathematically?

The Monty Hall Problem Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? What is the probability that you chose the car on the first try? the goat?

Activity Instructions: Get into pairs- choose who will roll the die and who will be choosing the cup the player turns around, can't see the cups Roll the die and quietly place the candy under the matching cup—Remember where ask the player to turn back around, and choose a cup, placing a finger on it lift one cup that that was not chosen, and that is empty Ask: "Stay or switch?" The player answers. show the cup that has the die write W or L in the chosen column, for Win or Lose Repeat up to 15 times as time allows!

Activity Instructions: How many marks (both Ws and Ls) in the Stay column? How many Ws in the Stay column? How many marks (both Ws and Ls) in the Switch column? How many Ws in the Switch column? Class Results??? In all, the instances in which switching yields winning results will be roughly twice as many as the instances in which staying gives a win.

The Monty Hall Problem In the original problem, what was your initial chance of winning the car when you picked your first door? ½ ⅔ ⅓ ¾ Answer: ⅓

The Monty Hall Problem In the original problem, what was your initial chance of picking the goat when you picked your first door? ½ ⅔ ⅓ ¾ Answer: ⅔

The Monty Hall Problem Monty Hall reveals a goat behind one of the doors. Staying with your original door gives you a 50 percent chance of winning the car. True False Answer: False

The Monty Hall Problem Why is it better for the contestant to switch doors? Why does this improve the contestant's chance of winning the car?

Scholarly Analysis In the early 1990s, Donald Granberg, Ph.D., University of Missouri, Center for Research in Social Behavior carried out a series of experiments with students, to investigate the psychology of Monty Hall choices. Students, at the University of Missouri, picked one of 3 doors, and each was then given an opportunity to switch. But, 174 of 190 (90%) decided not to switch. Likewise, 228 other college students were given the question on paper: only 13% switched. People tend to stick with their initial hunch, even though they ought to switch. "The implication is that the human brain is not wired to decipher readily the rational solution in the Monty Hall …. dilemmas." Why do most people get the answer wrong?

Think About It... Furthermore, Granberg asked, why do people stick even if they incorrectly assess the probabilities of winning by sticking or switching as 50 percent? How will I feel if I switch and lose? Is the affect greater when the situation involves action rather than inaction? People may feel worse, and therefore more likely to recall, when they change an answer on a multiple choice exam and it turns out to be incorrect than when they stick with a doubtful answer and it turns out to be incorrect.

Think About It... On Granberg's questionnaire, we found that people on the switch and lose condition anticipated they would feel more frustrated and angry than people in the stick and lose condition. Belief perseverance: a psychological phenomenon in which there is a tendency to persist with one’s held belief despite the fact that the information is inaccurate or that evidence shows otherwise Illusion of Control: people's belief that they have influence over the outcome of uncontrollable events