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Addressing Students’ Misconceptions about Probability. Leonid Khazanov. 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

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Presentation Transcript

Leonid Khazanov

• 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

• 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

• Misconceptions compete with normative concepts

• Students’ reasoning about probability is inconsistent

• Students lack in confidence when applying rules of probability

• #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

• Interviews

• Probability inventories

• Test instruments

• 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

• #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.

• #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.

• #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.

• #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.

• #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.

• 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

• 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?

• 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.

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

• The probability of no matches is

• Thus, P (at least one match)

1-

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%