STATS 10 Lecture Notes - Lecture 7: Random Number Table, Fair Coin, Face Card
Chapter 5: Probability
April 23, 25
SIMULATING RANDOMNESS
● randomness
: a situation where we know what outcomes could happen, but we don’t
know which particular outcome did or will happen
○ however, we can calculate the probability with which each outcome will happen
○ people are not good at identifying truly random samples or random experiments
→ so, need to rely on outside mechanisms
● real randomness is hard to achieve without a computer or some other device that
produces a truly random result
○ coin flips (to generate random 0’s and 1’s)
○ pick card to generate random numbers
○ pick number out of a hat
○ random number tables
● Example 1
: Simulate rolling a die 10 times using the following random number table.
○ Pick any line to start → let’s say we start with line 30
○ Go through the numbers, picking numbers 1-6 (and ignoring 0, 7, 8, 9)
○ The “random” numbers are: 5, 4, 5, 3, 4, 6, 2, 2, 5, 3
● Example 2
: (16% of Cars Fail Pollution Tests in California) → We can represent this
in many ways:
○ In a bag of 100 chips, 16 are red (represent cars that fail pollution tests) and the
remaining 84 are black (represent cars that pass the tests).
○ In a bag with 50 marbles, 8 are orange (represent cars that fail pollution tests)
and the remaining 42 are blue (represent cars that pass the tests).
●Example 3
: (16% of Cars Fail Pollution Tests in California) On a random number
table, numbers 00-15 represent cars that fail pollution tests and 16-99 represent cars
that pass the tests.
○ We would like to estimate the probability that an entire fleet of 7 cars would pass
using a random number table (assuming that each car is independent).
■ Start at row 1 of the random number table and read 2 digits at a time.
● if the number is between 00-15 → car will fail the pollution test
● if the number is between 16-99 → car will pass the test
■ A fleet of cars is comprised of 7 cars (ie. seven 2-digit random numbers)
→ if all seven cars pass the test, record a 1 (if not, record a 0)
■ Repeat many times and calculate the proportion of 1’s among the total
number of runs
● (ie. number of fleets where all cars passed the pollution test)
○ Run 1:
○ Run 2:
○ Run 3:
1 car failed → 11
○ Run 4:
2 cars failed → 07, 14
○ What proportion of runs got outcome “1”?
=total # runs
outcome "1" =4
1
○The average number of cars that fail the pollution test per run is
:
=total # runs
0 + 1 + 1 + 2 =4
4= 1
● *NOTES:
○ 4 runs = usually not considered sufficient
■ (for this class, use at least 5 runs)
○ You can start at any row you like on the random number table, but you should
make sure to note it when you’re writing up your simulation scheme.
○ You should not arbitrarily pick numbers from the random number table. → just
pick a row and follow across
■ otherwise, the number you’re using won’t be random (they will instead be
your choices)
PROBABILITY
● probability
: used to measure how often random events occur
○ theoretical probability
: the relative frequency at which an event happens after
infinitely many repetitions
■ ex. We may not be able to predict which song we play each time we hit
shuffle, however we know in the long-run 20 out of 1000 songs will be
Beyonce songs
○ empirical probability
: the relative frequency based on an experiment or on
observations of a real-life process
■ ex. We listened to 100 songs on shuffle and 4 of them are Beyonce
songs. The empirical probability is 4/100 = 0.04.
● theoretical probabilities are very abstract and can take forever to compute
○ hence, we can use empirical probabilities to estimate and test theoretical
probabilities
○ simulations
- experiments used to produce empirical probabilities
● for any random phenomenon, each attempt is called a trial
(and each trial generates an
outcome
)
○ trial = each time you hit shuffle
○ outcome = the song that plays as a result of hitting shuffle
● sample space
: the collection of all possible outcomes of a trial
○ sample space = the entire iTunes library of 1000 songs
● event
: a collection of outcomes
○ event = playing 2 Beyonce songs in a row in shuffle
● Calculating Probability: This is only true for equally likely outcomes.
(A)P=# of outcomes in A
# of all possible outcomes
Document Summary
Simulating randomness know which particular outcome did or will happen. Randomness : a situation where we know what outcomes could happen, but we don"t. However, we can calculate the probability with which each outcome will happen. People are not good at identifying truly random samples or random experiments. So, need to rely on outside mechanisms. Real randomness is hard to achieve without a computer or some other device that produces a truly random result. Coin flips (to generate random 0"s and 1"s) Example 1 : simulate rolling a die 10 times using the following random number table. Pick any line to start let"s say we start with line 30. Go through the numbers, picking numbers 1-6 (and ignoring 0, 7, 8, 9) The random numbers are: 5, 4, 5, 3, 4, 6, 2, 2, 5, 3. Example 2 : (16% of cars fail pollution tests in california) we can represent this.
