effects of sequence order on probabilistic reasoning

Effects of Sequence Order on

Probabilistic Reasoning

Jordan Baxter, Tori Edwards, Anna Morrow, Alex May and Eunice Makunzva

RandomnessBurns and Corpus (2004) deconstructed the concept of randomness to three fundamental principles:

● a fixed set of alternatives

● potential and prior outcome autonomy

● unbiased selection procedure (equiprobability)

Perceived violation of these principles within a situation that requires a choice between independent outcomes of

varying probability would dictate that the event was non-random as illustrated by the perception of streaks

● Streaks: unbroken continuity of the same outcome for at least three occurrences,

● Challenges:

o Law of small numbers, the belief that every segment of a random sequence should represent the total

probability of outcomes

o Burns and Corpus (2004)’s principles of equiprobability and outcome autonomy

● statistically inevitable

The violation of outcome autonomy can produce positive recency which would support the continuation of the

streak or negative recency which would define the streak as negligible, and predict the opposite of the last

outcome

● hot hand fallacy, the incorrect expectation that a streak will continue in a random sequence

● gambler’s fallacy, the incorrect expectation that streaks within random sequences will be balanced by the

occurrence of opposite outcomes B

Representativeness Heuristic (Law of Small

Numbers)

The expectation that a local sample will ultimately resemble the parent population● The emergence of streaks would be considered rare as it diverges from equal representation of the parent

population, and thus perceived as non-random (Sun & Wang, 2010)

The pivotal argument against this theory is its lack of definition: it proposes that both processes arise from the same

patterns of data (streaks), but it is unable distinguish the variables that inhibit/promote their separate expressions within

these data sets (Ayton & Fischer, 2004; Burns & Corpus, 2004; Sun & Wang, 2010).

Ayton & Fischer (2004)The hot hand fallacy and the gambler’s fallacy: Two faces of subjective

randomness?

● Explored the potential relationship between recency and the classification of differing sequence generators

o Interpretation of streaks generated from human performance are more likely to trigger the hot hand fallacy

(Ayton & Fischer, 2004).

Exhibited in golf putting, dart throwing, and signal detection, positive and negative streaks have

been produced following either success or failure due to the potential increase/decrease of

confidence after each attempt (Gilden & Wilson, 1995, as cited in Ayton & Fischer, 2004).

o It’s harder to perceive that inanimate objects can produce positively correlated sequences

Lack the characteristics of fatigue, confidence, and motivation which are seen as correlated within

human performance

● Examined probabilistic reasoning in determining the next outcome of a roulette sequence as well as their

confidence in their decision

o Confirmed their hypothesis: roulette predictions showed negative recency while confidence levels

portrayed positive recency

o Simultaneous: indistinguishable random sequences - at the same time - in the same head

Burns and Corpus (2004)Randomness and inductions from streaks: “Gambler’s fallacy” versus “hot

hand”● Proposed that individuals are more likely to exhibit the hot hand fallacy when interpreting perceived non-random streaks in

contrast to perceived random streaks which promote the gambler’s fallacy

● Manipulated the situational randomness of three constructed scenarios, and then presented participants with a scenario

specific probabilistic reasoning task concerning streak continuation.

o Random event - roulette wheel

o Non-random (Moderate) - Non-competition

o Non-random (Substantial) - Competition

● Burns & Corpus (2004) found that streaks were continued most often in the competitive scenario followed by non-

competitive, and then random, and conversely, the random scenario was more likely to induce the gambler’s fallacy.

● Ultimately, the study supported the proposed hypothesis that the perceived randomness of a sequence generator may

contribute to whether the hot hand or the gambler’s fallacy process is activated

Barron & Leider (2010)

Proposed that there is a crucial distinction between “Step by Step” and “End-of-Sequence” information processing

which can significantly affect the display of recency (positive or negative)

● Step-by-Step: presented an outcome one at a time to eventually complete the full sequence (recency as a result

of salience)

● End-of-Sequence: presented the entire sequence at once (no recency)

Three experimental conditions:

● Sequential condition: predicted the sequence of 11 roulette outcomes one at a time

● Simultaneous condition: predicted the 11th roulette outcome after being shown the previous 10 all at once

● Autosequential: predicted the 11th roulette outcome after watching the previous 10 revealed one at a time

Results supported their hypothesis: the Simultaneous conditioned showed the least amount of negative recency

(Gambler’s Fallacy) than those in the sequential and autosequential conditions

● Sequential presentation is an important antecedent for the gambler’s fallacy

● Simultaneous presentation of outcomes was found to attenuate the fallacy

Ball (2012)Not all streaks are the same: Individual differences in risk preferences during

runs of gains and losses

● Suggests that financial status will affect risk preferences, but streaks of gains and losses will still influence risk

preferences independently of the effects of status

○ The hot hand effect was more evident due to increased participant involvement in the choices of risk level

of success and the probability of the outcome of heads or tails

○ This circles back to Ayton and Fischer’s argument on positive recency: evidence that the hot hand effect

increases when the decision task has more human involvement than the effect when there is solely non

human mechanisms

Methods

Participants

With a total of 55 participants, 27 were males and 28 were females. All of the participants were

between the ages of 18 and 53, M=21.76 (SD=5.88). Participants were volunteers and no form of

compensation was offered in return for their participation.

Design

● One independent variable multiple level design with a between subjects manipulation

Materials

● Randomized number table

● One of three vignettes corresponding to assigned randomized number

● Writing utensil

Methods Continued

Casey, a student at the University of North Carolina Wilmington, has just completed his final

class of the day, and is now driving home. Construction on his usual route home has forced Casey to

take a detour, and now he has to use an alternative route of similar distance. He is not familiar with

this neighborhood, but is confident in his ability to navigate Wilmington and proceeds as normal. On

his way home, he hits three red lights in a row. As Casey continues on his way home, he sees a traffic

light in the distance.

1 2 3 4

Definitely Not Probably Not Probably Will Definitely Will

1. How likely is it that the next traffic light will be green?

2. How likely is it that the next traffic light will be red?

3. How likely is it that Casey will feel he is going to be later than usual arriving home?

4. How likely is it that Casey will feel that he will be earlier than usual arriving home?

Methods Ctd.

● The other two vignettes were identical with the exception of the nature of the traffic light

sequence. The second condition Casey hits three green lights and the third he hits a mixture of

the two.

● The four questions asked with each vignette were followed with ratings on a four point Lykert

scale

Procedure

● Experimenters collected responses, debriefed and thanked participants

● The experimenters pooled the data and entered it into excel, then processed through SPSS

In correspondence with the Gambler’s

Fallacy

Hypotheses:

1) If the previous light sequence was consecutively red,

we hypothesized the perceived future light color would

be green.

1) If the previous light sequence was consecutively green,

we hypothesized the perceived future light color would

be red.

Results

A One-Way ANOVA was conducted on each dependent measure

(Questions 1-4), and a post-hoc Tukey test

Examine if there was an effect of previous traffic light sequence on

perceived future sequence.

See Figure 1 for a depiction of one of the dependent variables

(Question 3) across all conditions.

Question 1● “How likely is it that the next traffic light will be green?”

● No main effect for light sequence F(2,52)=0.12; p>.05,

● So, no significant difference was found between the three traffic light sequence conditions

● Participants did not perceive the previously experienced traffic light sequence to have an effect on the probability

of hitting a green light next.

Mean SD n

Red Lights 2.40 .82 20

Green Lights 2.53 .74 15

Mixed Lights 2.45 .83 20

Note. SD=Standard Deviation. n=number of participants in each study

Table 1

Descriptive statistics for Question 1: How likely is it that the next light will be green?

Question 2● “How likely is it that the next traffic light will be red?”

● Again, no main effect for light sequence F(2, 52)=0.12; p>.05,

● So, no significant difference was found between the three light sequence conditions.

● Participants did not perceive the previously experienced light sequence to have an effect on the probability of

hitting a red light next.

Table 2

Descriptive statistics for Question 2: How likely is it that the next light will be red?

Mean SD n





Question 3● “How likely is it that Casey will feel he is going to be later than usual arriving home?”

● Sole dependent measure in the study to yield a main effect for light sequence F( 2,52)=4.88; p<.05.

● A post-hoc Tukey test was conducted:

o Condition 2 (M=2.27, SD= 1.03) and condition 3 ( M=3.15, SD=0.59) differed significantly at p<.05.

o Condition 1 (M=2.85, SD=0.88) was not significantly different

● Participants perceived that Casey would likely be late if he had previously hit a mixed light sequence more so than if he had hit a streak

of green lights, and participants who read the vignette with a streak of red lights did not find a significant difference either way.

Mean SD n


Green Lights 2.27 1.03 15


Table 3

Descriptive statistics of Question 3: How likely is it that Casey will feel he is going to be later

than usual arriving home?


Figure 1. Probability of Casey feeling that he’ll arrive home later than usual as

inquired in question three. A main effect was observed; specifically, a significant

difference was found between the Green and Mixed traffic light sequence

conditions following Tukey HSD and LSD post hoc tests. Means with differing

subscripts are significantly different, p < .05.

Question 4● “How likely is it that Casey will feel that he will be earlier than usual arriving home?”

● No main effect for light sequence F(2,52)=1.99; p>.05

● So, no significant difference was found between the three light conditions.

● Participants did not perceive the past sequence to have an effect on whether or not Casey feels he will be early arriving

home.

Mean SD n

Red Lights 2.15 1.09 20



Table 4

Descriptive statistics for Question 4: How likely is it that Casey will feel that he will be earlier than

usual arriving home?


Discussion

● Compared to Ayton & Fischer (2004)

o run length was significant, the longer the run the

more likely it is for subject to pick opposite

(gambler’s fallacy), inanimate streak

o confidence was increased by successful runs,

increased subjects’ confidence in prediction (hot

hand effect), human control streak

o Different results than current study for gambler’s

fallacy, found partial proof of hot hand effect

Discussion

● Compared to Ball (2012)

o Results are consistent with the gambler’s fallacy

o Results also are consistent with the hot hand effect

o Runs of gains/losses affect risky choices

o These results are inconsistent with our study

Discussion

● Compared with Burns & Corpus (2004)

o Found that participants were more likely to continue

a streak, but this was not the case for random

scenarios.

o These were not consistent with our results.

Discussion

● Compared to Barron (2010)

o HOW information is presented affects an individual’s

prediction

o The longer the run the more likely stronger

opposites will occur (Gambler’s Fallacy)

o Gambler’s Fallacy was not seen for simultaneous

condition which coincides with our findings

Limitations

● Sample size

● Convenience sample

● Untested vignettes

● Past driving experience

● Word choice in the vignettes

● Participant ability to see relation between lateness and

red light, earliness and green light

Future

● Implement risk

● Increase human participation

● Measure confidence

● Measure impulsiveness

● Simultaneous condition versus sequential condition

● Look at gambler’s fallacy and inanimate events versus

hot hand fallacy and human performance

References

Ayton, P., & Fischer, I. (2004). The hot hand fallacy and the gambler's fallacy: Two

faces of subjective randomness? Psychonomic Society, 32, 1369-1378. doi: 10.3758/BF03206327

Ball, C. T. (2012). Not all streaks are the same: Individual differences in risk preferences

during runs of gains and losses. Society for Judgment and Decision Making, 7, 452-461.

Barron, G., & Leider, S. (2010). The role of experience in the Gambler’s Fallacy. John

Wiley & Sons, 23, 117-129. doi: 10.1002/bdm.676

Burns, B., & Corpus B. (2004). Randomness and inductions from streaks: 'Gambler's

fallacy' versus 'hot hand'. Psychonomic Society, 11, 179-184. doi: 10.3758/BF03206480

Sun, Y., & Wang H. (2010). Perception of randomness: On the time of streaks. Elsevier

Science, 61, 333-342. doi:10.1016/j.cogpsych.2010.07.001

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effects of sequence order on probabilistic reasoning

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