flipping the dicejeremy orlo , jonathan bloom (mit math) flipping the dice june 23, 2014 11 / 19...

Flipping the Dice

Jeremy Orloff and Jonathan Bloom

Mathematics Department, MIT

[email protected] [email protected]

June 23, 2014

Jeremy Orloff, Jonathan Bloom (MIT Math) Flipping the Dice June 23, 2014 1 / 19

Bayesian dice demo

We’ve placed the five Platonic dice shown in a cup.

We need a volunteer to pick one at random.

What if the cup contained one thousand 20-sided dice?

Goal: make our intuition/learning mathematically precise.


Review of probability

Find the probability of each of the following events, assumingstandard (6-sided) dice are used.

Roll one die:1 Roll a 12 Roll a 43 Roll a 7

Roll two dice:1 Sum of two rolls is 42 Sum of two rolls is 7

We write p(roll 1) for the “probability of rolling a 1”.


Conditional probability

If we have many types of dice and we want to make it clear which onewe picked to roll, we write

p(roll 1 | pick 6-sided)

to mean “probability of rolling a 1 given that I picked a 6-sided die.”

Find:1 p(roll 1 | pick 4-sided)2 p(roll 5 | pick 4-sided)3 p(roll 5 | pick 6-sided)4 p(roll 5 | pick 20-sided)


Multiplication Rule

p(pick 6-sided and roll 1) = p(pick 6-sided) p(roll 1 | pick 6-sided)

P(A and B) = P(B)P(A|B)

Example. Suppose you have 2 four-sided dice and 3 six-sided dice.

Pick 6-sided die

Roll a 1 with 6-sided die

p(pick 6) = 3/5

(fraction of time you pick 6)

p(roll 1 |pick 6) = 1/6

(fraction of time you roll a 1given a 6-sided die)

Stage 1: pick a die

Stage 2: roll the die


The Law of Total Probability

Simplified notation:

P6 = pick a 6-sided die

R1 = roll a 1

p(R1 |P6) = probability we roll 1 given that we picked 6-sided die

Example. Suppose you have two 4-sided dice and three 6-sided dice.If I randomly pick a 4- or 6-sided die, then

p(R1) = p(P4 and R1) + p(P6 and R1)

= p(P4)p(R1 |P4) + p(P6)p(R1 |P6)

=2

5· 1

4+

3

5· 1

6= 0.2


Probability trees

Example. Have two 4-sided, three 6-sided. Find P(R1) and P(R5).

P4 P6

R1 R2 R3 R4 R1 R2 R3 R4 R5 R6

2/5 3/5

14

14

14

14

16

16

16

16

16

16

Pick die

Roll die

P(R1) = p(P4)p(R1 |P4) + p(P6)p(R1 |P6) =2

5· 1

4+

3

5· 1

6= 0.2

P(R5) = p(P4)p(R5 |P4) + p(P6)p(R5 |P6) =2

5· 0 +

3

5· 1

6= 0.1


Board question: probability trees

Before: cup contained two 4-sided dice and three 6-sided dice.

Now: cup contains three 4-sided dice and two 6-sided dice.

1 Is p(roll 1) now bigger, smaller, or the same as before?2 Make a tree.3 Find p(pick 6-sided and roll 1).4 Find p(roll 1).


Bayes theorem

The cup still contains three 4-sided dice and two 6-sided dice.

Suppose I pick a die and roll a 1.

What is the probability that I picked a 6-sided die?

We can find the answer using Bayes Theorem:

p(pick 6 | roll 1) =p(roll 1 | pick 6) p(pick 6)

p(roll 1)

or

p(P6 |R1) =p(R1 |P6) p(P6)

p(R1).


Bayesian updating

The cup still contains three 4-sided dice and two 6-sided dice.

I pick one at random and roll a 1.

What is the probability I picked a 4-sided die? a 6-sided die?That is, compute p(P4 |R1) and p(P6 |R1).

unnormalizedhypothesis prior likelihood posterior posteriorH p(H) p(R1|H) p(R1|H)p(H) p(H|R1)P4 3/5 1/4 3/20 9/13P6 2/5 1/6 2/30 4/13

total 1 p(R1) = 13/60 1

P4 P6

R1 R2 R3 R4 R1 R2 R3 R4 R5 R6

3/5 2/5

14

14

14

14

16

16

16

16

16

16

Pick die

Roll die


Board question: Bayesian updating

Now the cup contains one each of the 4, 6, 8, 12, and 20-sided dice.

We choose a die at random.

Question 1. Suppose we roll once and get a 7. What is the posteriorprobability of each die?

Question 2. Suppose we roll the same die a second time and get a9. Now what is the posterior probability of each die?


Applications

1 Medical testing2 Trial evidence3 Machine learning

e.g., classify e-mail as work, home or spam:

p(spam |word) =p(word | spam) p(spam)

p(word)

4 Genetic sequencinge.g., determine A, C, G, or T using sequencer data:

p(nucleotide | data) =p(data | nucleotide) p(nucleotide)

p(data)


18.05: Introduction to Probability and Statistics

Non-math majors, mostly life science and pre-med.For many, first and last course in statistics.

New curriculum

New pedagogy

New classroom

New technology


Active learning, flipped classroom

Meet 3 x 80min in TEAL room

60 students, 2 teachers, 3 assistants

Reading / reading questions on MITx

Minimal lecturing

Group problem solving at boards

Whole class and table discussions

Clicker questions

Computer-based studio using R

Traditional psets and pset checker


Active learning versus traditional lecture

Standing up is beneficial

Physical space is critical

Peer and teacher instruction

Student self-assessment

Teacher formative assessment


Technology and flipped classroom

Reading questions

Clickers and attendance

Pset checker

How much work was all this?

A tremendous amount because we changed so many things atonce.

How much are you able to cover?

More material with greater understanding.


Other observations

Active learning is more fun

Co-teaching is more fun

Students like getting to know their teachers

Students like targeted reading more than lecture video

Students love the pset checker


Course Arc

Pure probability

I Counting, random variables, distributions, quantiles, mean, varianceI Conditional probability, Bayes theorem, base rate fallacyI Joint distributions, covariance, correlation, independence

Pure applied probability (Statistics I)

I Bayesian inference with known priorsI Conjugate priors, probability intervals

Applied probability (Statistics II)

I Bayesian inference with unknown priorsI Frequentist significance tests and confidence intervalsI Linear regression, bootstrappingI Discussion of scientific papers

Computation, simulation and visualization using R and applets.


flipping the dicejeremy orlo , jonathan bloom (mit math) flipping the dice june 23, 2014 11 / 19...

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