Ch 17: Probability Models

1
Ch 17 Probability Models

• In this chapter we will introduce and work with 4
  different probability models.

2
Bernoulli Trials

• The basis for all 4 probability models we examine
  in this chapter is the Bernoulli trial.
• We have Bernoulli trials if
• there are only two possible outcomes (success and
  failure).
• the probability of success, p, is constant over
  all trials.
• the trials are independent.

3
The Geometric Model

• The Geometric probability model tells us the
  probability for a random variable that counts the
  number of Bernoulli trials until the first
  success.
• Geometric models Geom(p), are completely
  specified by one parameter, p, the probability of
  success
• p probability of success
• q 1 p probability of failure
• X of trials until the first success occurs
• P(X x) qx-1p
• We can also calculate the mean (expected of
  trials until success) and standard deviation.

4
Example Nuts over Nuts

• You get a job at the local chocolate factory
  packing boxes of mixed chocolates. Chocolates
  are stored in a huge bin (with thousands of
  chocolates) and look identical, but there are 5
  different flavors. 30 of the candies are solid,
  15 are filled with caramel, 25 have a
  butter-cream filling, 10 have a fruit filling
  and the rest are filled with nuts. One of the
  perks of your job is that you can sample their
  products on the job.
• If you pick one from the bin, whats the chance
  it is nut-filled?
• Whats the chance youd have to pick 4 chocolates
  before finding one that is nut-filled?
• How many chocolates would you expect to have to
  pick before finding a nut-filled one?
• What is the standard deviation for this picking
  example?

5
Another Geometric Example

• Remember the kid example from the last chapter
  (Ch 16, 5)? We will modify it slightly. A
  couple plans to have children until they get a
  girl, no matter how many children it takes.
  Again, we assume they are fertile and dont have
  twins.
• Create a probability model for the number of
  children that they will have. (it might also
  help to draw a tree diagram)
• Find the expected of children
• Does this differ from the expected of children
  when we had a cap of 3 kids?
• Whats the chance the couple will have to have
  more than 3 kids to get that daughter? More
  than 5 kids?
• Note- geometric models are usually easier to work
  with if you leave in fractional form

6
Independence

• One of the important requirements for Bernoulli
  trials is that the trials must be independent.
• When we dont have an infinite population, the
  trials are not actually independent. But there is
  a rule that allows us to pretend we have
  independent trials
• The 10 condition Bernoulli trials must be
  independent. If that assumption is violated
  because of a finite population considerations, it
  is still okay to proceed as long as the sample is
  smaller than 10 of the population.
• In the chocolate example, we could use the
  geometric because we were picking from a large
  bin. But what if we were picking from a box of
  10 chocolates where we knew 20 of the chocolates
  were nut-filled Could we still use the
  geometric calculate the probability that of
  needing to finding a nut filled chocolate if we
  grab 4 chocolates from the box?

7
Limitations of the Geometric Model

• The geometric model is only useful for situations
  where we experience failure until we find
  success.
• What if we want to figure out something more
  general, like
• Whats the chance that exactly 1 chocolate in a
  group of 5 is nut filled?
• Why cant we use the geometric for this?
• How many different ways could we select
  chocolates to meet this condition?
• How many different ways could we meet the
  condition that 2 chocolates in a group of 5 are
  nut filled?

8
Examples of Combinations

• To have 1 nut-filled chocolate in a group of 5
• Nxxxx, xNxxx, xxNxx, xxxNx, xxxxN
• 5 different ways!
• To have 2 nut-filled chocolates in a group of 5
• NNxxx, xNNxx, xxNNx, xxxNN,
• NxNxx, xNxNx, xxNxN,
• NxxNx, xNxxN,
• NxxxN,
• 10 different ways!
• Luckily, theres an easier way to keep track of
  this

9
The Binomial Model

• A Binomial model tells us the probability for a
  random variable that counts the number of
  successes in a fixed number of Bernoulli trials.
• Two parameters define the Binomial model n, the
  number of trials and, p, the probability of
  success. We denote this Binom(n, p).

10
The Binomial Model Combinations

• In n trials, there are
• ways to have k successes.
• Read nCk as n choose k.
• n! n x (n-1) x x 2 x 1, and n! is read as n
  factorial.
• 0! 1
• In Excel, you can use COMBIN(N,K). You can also
  use Excel compute the binomial, as shown in the
  textbook.
• The combination is how we account for the fact
  that there are multiple ways to get k successes.
• This calculation gets tough with large N.
• There will not be large combinations to calculate
  on my tests.

11
The Binomial Model (cont.)

• Binomial probability model for Bernoulli trials
• Binom(n,p)
• n number of trials
• p probability of success
• q 1 p probability of failure
• X of successes in n trials
• P(X x) nCx px qn-x

12
Example More Nuts over Nuts

• Given the same setup from before, you fill a mini
  box with 6 chocolates. Whats the chance that 2
  of the chocolates are nut-filled?
• What distribution do we need to use, binomial or
  geometric?
• Compute the probability of finding exactly 2
  nut-filled chocolates in a box of 6 chocolates.
• Compute the expected value and standard deviation
  for the number of nut-filled chocolate in a box
  of 6.
• What do we do with fractional answers? What do
  they mean?

13
The Normal Model to the Rescue!

• When dealing with a large number of trials in a
  Binomial situation, making direct calculations of
  the probabilities becomes tedious (or outright
  impossible).
• Fortunately, as long as the Success/Failure
  Condition holds, we can use the Normal model to
  approximate Binomial probabilities.
• The normal uses the same parameters for the mean
  and standard deviation m np and
• Be sure to check the Success/failure condition A
  Binomial model can be considered approximately
  Normal if we expect at least 10 successes and 10
  failures in our trials
• np 10 and nq 10

14
Continuous Random Variables

• When we use the Normal model to approximate the
  Binomial model, we are using a continuous random
  variable to approximate a discrete random
  variable.
• Warning With continuous variables we need to
  work with intervals, as the chance that a number
  matches exactly is 0.
• i.e. The probability that someone is exactly 64
  tall on a continuous scale is the same as saying
  they are 64.0000 tall, not 63.9999 or 64.0001
  tall. We might instead mean Greater than
  63.5, less than 64.5
• So, when we use the Normal model, we no longer
  calculate the probability that the random
  variable equals a particular value, but only that
  it lies between two values (one of those values
  may be zero or infinity.)

15
Example Yet more Nuts

• Given the same setup from before, you now have to
  fill a mega-box with 100 chocolates. Whats the
  chance that fewer than 25 of the chocolates are
  nut-filled?
• Why would it be difficult to compute this as a
  binomial?
• What probability model can we use in place of the
  binomial? Justify why!
• Whats the expected number of nut-filled
  chocolates (and standard deviation) in the box
• Whats the chance the box has fewer than 25
  nut-filled chocolates?
• and the chance the box has at least 25
  nut-filled chocolates?
• Aside one of the problems with the normal
  approximation is that whether to interpret it as
  P(Xlt 25) or P(X lt 25). I will accept either
  interpretation.

16
One Last Caveat About Using a Normal Model to
Approximate a Binomial...

• Substituting the normal for the binomial will not
  give the exact same answers. (Theres often
  around a 2-3 difference between them).
• Problem 25 in the assigned problems works out
  the probabilities for both the Binomial model and
  its normal approximation.
• Technically the binomial is more accurate (the
  normal is only an approximation).
• But the normal model is easier to compute, and
  when conditions are satisfied, you can use it,
  even with this inaccuracy!
• And what can we do if np lt 10, but n is large?
  Does this mean we have to use the binomial?

17
The Poisson Model

• The Poisson probability model was originally
  derived to approximate the Binomial model when
  the probability of success, p, is very small and
  the number of trials, n, is very large.
• Rule of thumb 1 (not in book) You should have
  more than 20 trials and no more than a 5 chance
  of success for the Poisson.
• Rule of thumb 2 (also not in the book) In
  general, its a bad idea to use the binomial when
  n gt 100, because rounding errors will make most
  binomial computations unstable.
• The parameter for the Poisson model is ?. To
  approximate a Binomial model with a Poisson
  model, just make their means match ? np.
• The Poisson is a useful for looking at very rare
  events that have major consequences for example
  accidents, terrorist incidents, lottery
  winnings...
• It requires only that the events be independent
  and that the mean number of occurrences stays
  constant over time.

18
The Poisson Model (cont.)

• Poisson probability model for successes
  Poisson(?)
• ? mean number of successes np
• X of successes
• e is an important mathematical constant (
  2.71828)

19
Example Nuts over Prizes

• The chocolate company is running a promotion,
  where 0.1 of their chocolates are actually
  chocolate-covered rubber balls that can be
  redeemed for a cash prize (the company hops that
  consumers wont swallow their candies whole and
  choke!) Chris buys and eats 5 mega-boxes of
  chocolates. Assuming the prize chocolates are
  distributed randomly...
• What model are we likely to use? Why cant we
  use a Normal? Why dont we want to use a
  Binomial?
• What is the expected number of prizes (and SD)
• Compute the probability that Chris wins exactly 1
  prize.
• Compute the probability that Chris wins nothing.
• Compute the probability that Chris wins more than
  one prize (Hint theres an easy way to do
  this!)

20
What Can Go Wrong?

• Be sure you have Bernoulli trials.
• You need two outcomes per trial, a constant
  probability of success, and independence.
• Remember that the 10 Condition provides a
  reasonable substitute for independence.
• Dont confuse Geometric and Binomial models.
• Dont use either the Normal approximation or the
  Poisson with small n.
• You need an expectation of at least 10 successes
  and 10 failures to use the Normal approximation.
• Conversely, avoid using the binomial model for
  large n.

21
What have we learned?

• Bernoulli trials show up in a lot of places, and
  depending on the situation, can be represented
  with one of 4 models...
• Geometric model
• When were interested in the number of Bernoulli
  trials until the next success.
• Binomial model
• When were interested in the number of successes
  in a certain number of Bernoulli trials.
• Normal model
• To approximate a Binomial model when we expect at
  least 10 successes and 10 failures.
• Poisson model
• To approximate a Binomial model when there are a
  very large number of trials (certainly more than
  20!) and the probability of success (or failure)
  is very small.

22
When to Use Which Model A Guide

• All of these are based on Bernoulli Success or
  No Success events
• Geometric- Examines the question of how many
  times do we have to try until we succeed?
• Binomial- Examines multiple successes in a
  series of trials
• Normal- allows us to avoid the hassle of using
  the binomial for lots of trials if we expect to
  have at least several successes and several
  failures
• Poisson- allows us to avoid the hassle of using
  the binomial for lots of trials if we expect not
  to have many successes.

23
Example Bernoulli or Not?

• 1 Can we use probability models based on
  Bernoulli trials to investigate the following
  situations? Why or Why not? What further
  assumptions may be necessary?
• We roll 50 dice to find the distribution of the
  of spots on the faces.
• How likely is it that in a group of 120, the
  majority have type A blood, given that it is
  found in 43 of the population?
• We deal 5 cards from a standard deck and get all
  hearts- whats the likelihood of that?
• We pool 500 out of the 3000 potential voters to
  see if they favor the budget.
• A company realizes that 10 of its packages dont
  seal properly. Whats the chance that more than
  3 are defective in a pack of 24?

24
Problems

• 32 (with some additional questions). Suppose
  the probability of a major Bay Area earthquake on
  any given day is 1 out of 10,000.
• What distribution are we likely to use?
• Whats the expected number of major earthquakes
  in the next 1000 days?
• What is the probability there will be at least 1
  major earthquake in the next 1000 days?
• If the conditions for our probability model truly
  hold in reality, does the chance of a major
  earthquake in the next 1000 days change if we
  just had one today?

25
Example Old Exam Problem

• Luigi likes to ask ladies out. Every day he
  makes it a habit to ask an unknown attractive
  woman for her business card so he can call her
  (he has already figured out they will give him a
  fake number if he doesnt get proof!) Alas, most
  (90) of the time he is rejected. Still, he
  realizes if he asks enough, he will end up with a
  few cards.
• Find the probability he collects at least 1 card
  in 4 days
• He plans to continue this practice for the next
  250 days to see if he can get at least 28 cards.
• Can we use a normal to approximate this?
• Show why or why not (2 with calculations)
• Now find the probability

26
Example Another Exam Question

• Your professor is absentminded. For instance,
  she has a 20 chance of forgetting to close the
  garage door when she drives to work. She wants
  to determine what the probability is she will
  forget to close the garage door exactly twice
  during the 5 days that she drives to work next
  week. Assume that whether she forgets on one day
  will not effect other days. Calculate the
  probability that she forgets to close the garage
  door exactly twice in 5 days.
• Her absent mindedness leads her to forget where
  she left USB memory key, and she often has to
  search for it all over her office. Every place
  she looks, she has a 15 chance of finding it,
  and she will, of course, stop tearing her office
  apart once she finds it. Calculate the
  probability that she needs to look exactly 3
  places to find her key

27
Another Exam Question (Cont.)

• The absentminded professor is often distracted
  from paying attention to the road. Thus she has
  a 0.3 chance of being involved in a minor
  accident every day she commutes to school. If she
  has an accident one day, it will not affect her
  chances of having an accident on other days.
• A) Calculate the mean number of accidents per
  year given she commutes to work 200 days a year
• B) Could we use a Normal distribution to
  approximate this? Why or why not ?
• C) Calculate the chance that she has probability
  that she has exactly 1 accident that year
• D) Calculate the chance she has more than 1
  accident.