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9B. Random Simulations

Topics:

The class random

Estimating probabilities

Estimating averages

More occasions to practice iteration

The random Module

Contains functions that can be used

in the design of random simulations.

We will practice with these:

random.randint(a,b) random.uniform(a,b) random.normalvariate(mu,sigma) And as a fringe benefit, more practice with for-loops

Generating Random Integers

7OMP LV RH UMQGRPO\ VHOHŃP MQ HOHPHQP IURP POH VHP SMMĄ1"N` MQG MVVLJQ LP PR Q

If a and b are initialized integers with a < b

then i = random.randint(a,b) assigns to i M ´UMQGRPµ LQPHJHU POMP VMPLVILHV a <= i <= b

JOMP GRHV ´5MQGRPµ 0HMQ"

import random for k in range(1000000): i = random.randint(1,6) print i

7OH RXPSXP RRXOG ´ORRN OLNHµ \RX UROOHG M GLŃH

one million times and recorded the outcomes.

No discernible pattern.

5RXJOO\ HTXMO QXPNHUV RI 1·V 2·V 3·V 4·V D·V MQG 6·VB

Renaming Imported Functions

import random for k in range(1000000): i = random.randint(1,6) print i from random import randint as randi for k in range(1000000): i = randi(1,6) print i Handy when the names are long or when you just want to name things your way.

Random Simulation

We can use randint to simulate genuinely

random events, e.g.,

Flip a coin one million times and record the

number of heads and tails.

Coin Toss

from random import randint as randi

N = 1000000

Heads = 0

Tails = 0

for k in range(N): i = randi(1,2) if i==1:

Heads = Heads+1

else:

Tails = Tails+1

print N, Heads, Tails

7OH ´ŃRXQPµ YMULMNOHV Heads

and Tails are initialized randi returns 1 or 2

FRQYHQPLRQ ´1µ LV OHMGV

FRQYHQPLRQ ´2µ LV PMLOV

A Handy Short Cut

Incrementing a variable is such a common

calculation that Python supports a shortcut.

These are equivalent:

x += 1 x = x+1 x += c is equivalent to x = x+c

Coin Toss

from random import randint as randi

N = 1000000

Heads = 0

Tails = 0

for k in range(N): i = randi(1,2) if i==1:

Heads+=1

else:

Tails+=1

print N, Heads, Tails

7OH ´ŃRXQPµ YMULMNOHV Heads

and Tails are initialized randi returns 1 or 2

FRQYHQPLRQ ´1µ LV OHMGV

FRQYHQPLRQ ´2µ LV PMLOV

Sample Outputs

N = 1000000

Heads = 500636

Tails = 499364

N = 1000000

Heads = 499354

Tails = 500646

Different runs produce

different results.

This is consistent with

what would happen if we physically tossed a coin one million times.

Estimating Probabilities

You roll a dice. What is the probability

POMP POH RXPŃRPH LV ´Dµ"

Of course, we know the answer is 1/6. But

OHP·V ´GLVŃRYHUµ POLV POURXJO VLPXOMPLRQB

Dice Roll

from random import randint as randi

N = 6000000

count = 0 for k in range(N): i = randi(1,6) if i==5: count+=1 prob = float(count)/float(N) print prob

N is the number of

´H[SHULPHQPVµB

i is the outcome of an experiment prob is the probability the outcome is 5

Dice Roll

from random import randint as randi

N = 6000000

count = 0 for k in range(N): i = randi(1,6) if i==5: count+=1 prob = float(count)/float(N) print prob

Output:

.166837

Discovery Through Simulation

Roll three dice.

What is the probability that the three

outcomes are all different ?

If you know a little math, you can do this

RLPORXP POH ŃRPSXPHUB IHP·V MVVXPH POMP

RH GRQ·P NQRR POMP PMPOB

Solution

N = 1000000

count = 0 for k in range(1,N+1): d1 = randi(1,6) d2 = randi(1,6) d3 = randi(1,6) if d1!=d2 and d2!=d3 and d3!=d1: count +=1 if k%100000==0: print k,float(count)/float(k) Prints snapshots of the probability estimates every 100,000 trials

Note the

3 calls to

randi

Sample Output

k count/k

[PDF] [PDF] 9B Random Simulations - Cornell Computer Science

9B. Random Simulations

Topics:

The class random

Estimating probabilities

Estimating averages

More occasions to practice iteration

The random Module

Contains functions that can be used

We will practice with these:

Generating Random Integers

7OMP LV RH UMQGRPO\ VHOHŃP MQ HOHPHQP IURP POH VHP SMMĄ1"N` MQG MVVLJQ LP PR Q

If a and b are initialized integers with a < b

JOMP GRHV ´5MQGRPµ 0HMQ"

7OH RXPSXP RRXOG ´ORRN OLNHµ \RX UROOHG M GLŃH

No discernible pattern.

5RXJOO\ HTXMO QXPNHUV RI 1·V 2·V 3·V 4·V D·V MQG 6·VB

Renaming Imported Functions

Random Simulation

We can use randint to simulate genuinely

Flip a coin one million times and record the

Coin Toss

N = 1000000

Heads = 0

Tails = 0

Heads = Heads+1

Tails = Tails+1

7OH ´ŃRXQPµ YMULMNOHV Heads

FRQYHQPLRQ ´1µ LV OHMGV

FRQYHQPLRQ ´2µ LV PMLOV

A Handy Short Cut

Incrementing a variable is such a common

These are equivalent:

Coin Toss

N = 1000000

Heads = 0

Tails = 0

Heads+=1

Tails+=1

7OH ´ŃRXQPµ YMULMNOHV Heads

FRQYHQPLRQ ´1µ LV OHMGV

FRQYHQPLRQ ´2µ LV PMLOV

Sample Outputs

N = 1000000

Heads = 500636

Tails = 499364

N = 1000000

Heads = 499354

Tails = 500646

Different runs produce

This is consistent with

Estimating Probabilities

You roll a dice. What is the probability

POMP POH RXPŃRPH LV ´Dµ"

Of course, we know the answer is 1/6. But

Dice Roll

N = 6000000

N is the number of

´H[SHULPHQPVµB

Dice Roll

N = 6000000

Output:

Discovery Through Simulation

Roll three dice.

What is the probability that the three

If you know a little math, you can do this

RLPORXP POH ŃRPSXPHUB IHP·V MVVXPH POMP

RH GRQ·P NQRR POMP PMPOB

Solution

N = 1000000

Note the

3 calls to

Sample Output

100000 0.554080

200000 0.555125

300000 0.555443

400000 0.555512

500000 0.555882

600000 0.555750

700000 0.555901