First Problem Set: Answers, Discussion and Background Part I. Intuition Concerning Probability Do these problems individually Answer the following questions based upon your intuitive understanding about probability, without formal computation and before doing any readings. (Of course, logical thinking is allowed!) There are no right and wrong answers for the problems in this section and we will return to some of these later to do formal calculations. Tversky and Kahneman published their studies about probability biases in the 1970 s, and they have been extensively investigated since. 1. In 1991, approximately what proportion of people in the USA were victims of robbery, according to the official statistics. Actual Answer: 1/500. Common Answers: Almost everyone says it is much higher. BIAS: Ease of recall or ease of imagination causes over-assessment of the probability. 2. In flipping a fair coin, which of the following sequences is most likely to occur: 1st flip 2nd flip 3rd flip 4th flip 5th flip 6th flip (a) Head Tail Head Tail Head Tail (b) Head Head Head Tail Tail Tail (c) Head Head Head Head Head Head (d) Head Head Tail Head Tail Tail Actual Answer: All are equally likely (coin has no memory). Common Answers: Most think they are quite different. BIAS: Expecting sequences of events to appear random; but note that many more sequences do look similar to (d) than to the others. Let s write out the possibilities for 4 flips. 3. Susie has been driving for 40 years without having a road accident. A friend tells her that the chances of her being involved in an accident in the next five years must be high because the probability of an individual driving for 45 years without an accident is low. Is this thinking correct? Actual Answer: No. BIAS: Expecting chance to be self-correcting; but it has no memory
Another question: Would you expect Susie to have more or less of a chance of having an accident in the next 5 years than the typical individual? 5. (Sedlmeier, 1999) Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Please indicate which of the two following statements is more likely: Linda is a bank teller Linda is a bank teller and is active in the feminist movement. Actual Answer: Fact: P(A & B) < P(A) (called the conjunctive rule) for any events A and B, so P(Bank Teller & Active ) < P(Bank Teller). Think of the population being considered as the set of people satisfying the description; bank tellers are a subset of this population; feminist bank tellers are a subset of that population. Common Answers: 70% to 90% get this wrong! BIAS: Another conjunctive fallacy, perhaps based on natural language interpretations of statements (with the first statement implying the exclusion of being in the feminist movement). RELEVANT TEXT READING: Mutually Exclusive and Exhaustive Events The Addition Rule If the events are mutually exclusive then the addition rule is: P(A or B) = P(A) + P(B) (where A and B are the events) If the events are not mutually exclusive we should apply the addition rule as P(A or B) = P(A) +P(B) - P(A and B). Complementary Events: P(A) = 1 - P(A) Marginal and Conditional Probabilities P(B) is the marginal probability of B happening P(A&B) is the joint probability of A and B both happening P(A B) = P(A & B) / P(B) is the conditional probability of A happening, given that B has happened. 6. (Bar-Hillel and Falk, 1982). Three cards are in a hat. One is red on both sides, denoted RR. One is white on both sides, denoted WW. One is red on one side and white on the other, denoted RW. We draw one card blindly and put it on the table. It shows a Red face up, denoted Ru. What is the probability that the hidden side is also red? Actual Answer- 2/3: work out the state space answer for P(RR Ru), and the formula Common Answer: ½ (perhaps because only two possibilities are left) BIAS: Overestimating the probability of conjunctive events
7. (Sedlmeier, 1999) Some years ago, the Bavarian police concluded in a statistical survey that 60% of heroin addicts had smoked marijuana before they became heroin addicted. The Bavarian secretary of state regarded this as proof that marijuana is an entrance drug. If somebody smokes marijuana, he argued, he or she will later on end up (with a probability of about 60%) being a heroin addict. What do you think about this conclusion is the Bavarian secretary of state right? (Why or why not?) Actual Answer: No Common Answer: Yes BIAS: Assuming p(marijuana use heroin use) = p(heroin use marijuana use), possibly caused by thinking causally. (Imagine replacing marijuana by chocolate to see the error of this.) Do Joint Table. 8. (Tversky & Kahneman, 1982) In a survey of high-school seniors in a city, the height of boys was compared to the height of their fathers. In which prediction would you have greater confidence? (a) The prediction of the father's height from the son's height. (b) The prediction of the son's height from the father's height (c) Equal confidence Actual Answer: Equal confidence Common Answer: (b) BIAS: Erroneously thinking causally. Here p(father height son height) = p(son height father height), because marginal probabilities are the same. 4. An electronic safety system, which will automatically shut off machinery in the event of an emergency, is being proposed for a factory. It would consist of 150 independent components, each of which must work if the entire system is to be operational. On any day, each component would be designed to have a 99.5% probability of working. Estimate the probability that the entire safety system would be operational on any given day if a decision was made to install it. Actual Answer: P(operational)=(0.995) 150 = 0.47. Common Answers: Most think it is much larger. BIAS: Overestimating the probability of conjunctive (i.e., simultaneously occurring) events. RELEVANT TEXT READING Independent and Dependent Events Two events, A and B, are said to be independent if the probability of event A occurring is unaffected by the occurrence or non-occurrence of event B, i.e. P(A B) = P(A) The Multiplication Rule P(A and B) =P(A) P(B) if A and B are independent
9. The Monty Hall Problem: A contestant in the show is faced with three doors. She knows that behind one door there is a desirable prize (e.g., a fancy car) and behind the other two there are undesirable items (e.g. a goat). The contestant picks a door and Monty Hall says, "I will now reveal to you what is behind one of the other doors." One door is opened and there is a goat behind it. (Of course, Monty will only open a door with a goat behind it). Then Monty says "Would you like to change your door selection?" Should you change? Actual Answer: Switch Common Answer: It s irrelevant BIAS: Incorrect Conditioning. (Proving the right answer is a problem for next week.) 10. (Sedlmeier, 1999) A bookbag contains 1,000 poker chips. Suppose you start with two such bags, one containing 700 red and 300 blue chips, the other containing 300 red and 700 blue. I flipped a fair coin to determine which one to use. Thus, if your opinions are like mine, your probability at the moment that this is the predominantly red bookbag is 0.5. Now, you draw chips randomly from the bag, replacing the chip (and mixing up the bag) after each draw. In 12 draws, you get 8 reds and 4 blues. Now, on the basis of everything you know, what is your intuitive probability that this is the predominantly red bag? Actual Answer: 0.97 (to be done later as homework) Common Answer: between 0.7 and 0.8 BIAS: Incorrect assessment of the strength of evidence in data 11. (Kahneman and Tversky, 1972) A town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys. The exact percentage of baby boys, however, varies from day to day. Sometimes it may be higher than 50%, sometimes lower. For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days? (a) The larger hospital (b) The smaller hospital (c) About the same (i.e., within 5% of each other). Actual Answer: smaller hospital (see below) Common Answer: about 1/3 choose each answer BIAS: Incorrect assessment of the probability of deviation from the norm (see the relevant probability distributions at the end of these notes).
Part II. Problems to Formally Analyze You may work these problems with your study group, and after doing the readings. 1. A king of a small country has one sibling. Assuming the usual rules (that is, the oldest boy gets to be king), what is the probability that his sibling is a boy (prince)? Note that if the oldest child is a girl, and the youngest is a boy, then the younger child will be king. (a) First, make a guess at what the answer is. (b) Second, using two different coins (e.g., a quarter and a nickel), simulate the situation 100 times. Lets say that Heads is "boy" and Tails is "girl", and that the quarter represents the older child. Toss the two coins together and decide whether there is a king and if so, whether the other coin represents a prince or a princess. Tally the results. (c) Third, write down a complete set of possible outcomes for this problem. Assign a probability to each outcome. Then conditionalize on the fact that there is a king. Calculate the conditional probability that the sibling is a boy, given that there is a king. Does your result agree with what you got by simulation? (d) Finally, suppose the king has two siblings. What is the probability that he has exactly one brother? Two brothers? 2. Imagine rolling two ordinary dice. Write down a complete set of possible outcomes for the two dice. How many different outcomes are there? Assign each outcome a probability, assuming that the dice are fair. (a) What is the probability that you will roll "8" (total of both dice)?" (b) What is the probability that you will roll "8", given that one of the dice is a "5"? (c) What is the probability that one of the dice is a "5", given that you have rolled "8"? 3. Funny dice: You have two dice, A and B. Die A has four sides with four spots and two sides with no spots. Die B has three sides with five spots and three sides with 1 spot. If you toss the two dice together, what is the probability that Die A will beat Die B? 4. Imagine flipping a penny, a nickel and a dime together. (a) Write down a list of all the possible outcomes of this experiment. Assign each outcome a probability. (b) What is the probability that you will see three presidents? (c) What is the probability that you will see exactly two presidents? (d) What is the probability of seeing three presidents, given that at least one president is showing? (e) What is the probability of seeing three presidents, given that Lincoln is showing?
Part III. An Experiment Do this individually For class on Thursday, I ask each student (individually, not as a group) to take a coin and flip it. (I want 12 independent data sets, one for each of you!) If it is heads, then flip it 100 times more, recording the exact sequence of heads and tails. Count the number of heads and tails. Turn in both the exact sequence of heads and tails and the number of each. If it is tails, then try to make up a fake sequence of heads and tails that you think might have come from flipping a coin. Try to fool me by making your sequence as random looking as possible. Turn in both the sequence you invented and the total number of heads and tails in your sequence. Do not put any indication of whether your sequence is real or fake on your paper. We will use these numbers in class on Thursday. Probability Trees
Probability Distributions
Probability distributions for # boys born per day for the small and the large hospital, respectively (assuming there are 15 and 45 babies born in each, respectively).