1 Addressing Students’ Misconceptions about Probability Leonid Khazanov

2 Typical Misconceptions about Probability Representativeness: subjects estimate the likelihood of an event based on how well an outcome represents some aspect of the parent population Gambler’s fallacy: subjects tend to believe that after a run of heads, tails should be more likely to come up Equiprobability: Attributing the same probability to different events in a random experiment regardless of the chances in favor or against it

3 Typical Misconceptions Availability bias: subjects estimate the likelihood of events based on how easy it is for them to call in mind particular instances of the event. Conjunction fallacy: many subjects are prone to rate certain types of conjunctive events as much more likely to occur than their parent stem events. Outcome orientation: subjects do not see the results of a single trial as embedded in a sample of many such trials. They perceive each trial as a separate individual phenomenon

4 How is learning effected by misconceptions? Misconceptions compete with normative concepts Students’ reasoning about probability is inconsistent Students lack in confidence when applying rules of probability

5 Two Problems: Similar or Different? #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is MOST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally likely and both less likely than a e. All of the arrangements are equally likely #2. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely

6 Methods of assessing misconceptions Interviews Probability inventories Test instruments

7 Probability Reasoning Questionnaire Two-part multiple-choice questions: the principal question and justification Distracters consistent with at least two misconceptions on each item The context of problems varies from purely academic to real life The instrument is valid and reliable

8 Item from PRQ: correct response #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely. Which of the following best describes the reason for your answer to the preceding question? a. Impossible to tell which arrangement will not occur, any of the arrangements could fail to occur b. In this situation each of the ordered arrangements has the same probability of occurring or not occurring c. Tossing of a coin is random, and random events always have the same probability of occurring or not occurring d. There ought to be about the same number of heads and tails e. Since tossing of a coin is random, the coin is unlikely to alternate between heads and tails.

9 Representativeness #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely. Which of the following best describes the reason for your answer to the preceding question? a. Impossible to tell which arrangement will not occur, any of the arrangements could fail to occur b. In this situation each of the ordered arrangements has the same probability of occurring or not occurring c. Tossing of a coin is random, and random events always have the same probability of occurring or not occurring d. There ought to be about the same number of heads and tails e. Since tossing of a coin is random, the coin is unlikely to alternate between heads and tails.

10 More Representativeness #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely. Which of the following best describes the reason for your answer to the preceding question? a. Impossible to tell which arrangement will not occur, any of the arrangements could fail to occur b. In this situation each of the ordered arrangements has the same probability of occurring or not occurring c. Tossing of a coin is random, and random events always have the same probability of occurring or not occurring d. There ought to be about the same number of heads and tails e. Since tossing of a coin is random, the coin is unlikely to alternate between heads and tails.

11 Equiprobability Bias #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely. Which of the following best describes the reason for your answer to the preceding question? a. Impossible to tell which arrangement will not occur, any of the arrangements could fail to occur b. In this situation each of the ordered arrangements has the same probability of occurring or not occurring c. Tossing of a coin is random, and random events always have the same probability of occurring or not occurring d. There ought to be about the same number of heads and tails e. Since tossing of a coin is random, the coin is unlikely to alternate between heads and tails.

12 Outcome orientation #1. If a fair coin is tossed 6 times, which of the following ordered arrangements of heads and tails is LEAST likely to occur? a. HTTHTH b. HHHHHT c. HTHTHT d. b & c are equally unlikely and both less likely than a. e. All of the arrangements are equally unlikely. Which of the following best describes the reason for your answer to the preceding question? a. Impossible to tell which arrangement will not occur, any of the arrangements could fail to occur b. In this situation each of the ordered arrangements has the same probability of occurring or not occurring c. Tossing of a coin is random, and random events always have the same probability of occurring or not occurring d. There ought to be about the same number of heads and tails e. Since tossing of a coin is random, the coin is unlikely to alternate between heads and tails.

13 Discussion Situation Discussion situation 2. Best chance of winning. Misconception treated: reperesentativeness Link to important concepts: Law of large numbers, independence, binomial distribution Placement in the course: when discussing the law of large numbers Time: about 20 minutes Format: small groups or whole class discussion Problem statement: You finished first in a chess tournament. You are confident that you are indeed the best player. However, the rules require that you must compete in a playoff against the student who finished second in the tournament. What would you prefer: a) a 5-game series, b) a 9-game series?

14 What are the risks involved in the teaching of probability? HTHTHTHTHT Students often tend to assign lower probability to outcomes that look special. Saying that the above sequence is very rare may reinforce a misconception.

15 Matched Birthday Problem In a group of N people what is the probability that at least 2 of them will have their birthday on the same day?

16 Solution The probability of no matches is Thus, P (at least one match) 1-

17 What is the problem with this summary of probabilities? Number Probability Probability that of people that all birthdays there are at least two people in group are different who share the same birthday 2 100% 0% 4 98% 2% 6 96% 4% 8 93% 7% … … … 74 0% 100%