Transcript 7.2 Expected Value, gambler`s fallacy, and the law of large numbers

AP STATS:
Take 10 minutes or so to complete your 7.1C quiz.
When you finish, take a look at your quiz and your project and than
read the page from How to Lie with Statistics and we’ll examine a
recent ad related to this idea.
Some notes from the project
• Very nicely done and some creative project ideas as well.
• Make sure that you are explicit in how you conduct the
experiment. Again someone should be able to read the study
and replicate it.
• No calculator speak.
• Make sure that there is something quantitative that you are
measuring in order to determine whether your results were
significant.
Agenda
• We will be covering 7.2 today and tomorrow.
• We will review on Wednesday for the Chapter 6/7 Test.
• It will be half multiple choice and half free response.
• I will be putting up a lesson from an AP book online. It’s a
good idea to read this as it is a summary of Chapter 6 and 7
(but more condensed and with some examples).
Video of the Day!
How to Lie with Statistics?
http://www.ispot.tv/ad/75Zn/verizon-mapgallery
A few reminders….
• P(X>1)
• P(1<X<3)
• P(X ≠2)
• What does the area under the density curve always equal?
• Mean is the balance point.
• Median is the equal areas point
Reminders normal distribution
The mean of a random variable.
• The mean of a random variable (X) is a mean that is weighted
by the probability of each event occurring. It is a weighted
average.
• The mean of a probability distribution is denoted as mu-x.
• We oftentimes call this the EXPECTED VALUE. The amount
that we can expect to gain/lose ON AVERAGE in the long run,
on each “play” of the game. Originally developed for the use
of gambling (used frequently for things like insurance).
• It is the sum of x
Lottery Example
• Imagine the probability of winning the lotto is 1/1,000,000.
• The payout if you win is $800,000. Lottery tickets cost a $1.
What is the expected value of a lottery ticket?
Expected Value
Variance of a discrete random
variable
Example
Sampling Distributions
• One of the areas that this course is heading in is towards statistical
inference. In other words, estimating a population parameter (like the
population mean) from a sample. In other words, we will be
approximating μ using a sample mean (x-bar). (We will study sampling
distributions in Chapter 9.)
• Say we wanted to find the mean height of all women in the US aged
18-24. We might take an SRS of women and use their sample mean to
represent the true mean (μ) with a certain level of precision.
• The LAW of LARGE NUMBERS the mean of the sample will approach
the mean of the population (μ) as you increase the number of people
sampled!
** this holds for ANY population (not just normal
distributions)
Think flipping quarters.
Or how casinos make
money in the long run
even though they might
have a “bad” night.
A Psychological Shortcoming
• Most people believe that short sequences of random events will show
the same type of average behavior that we will only see in the long run.
Casinos thrive off of this as it keeps gamblers gambling.
• In other words, people falsely believe in a LAW OF SMALL NUMBERS
which does not exist.
• This has to do with the fallacy of the “hot hand” which I think we
discussed….
What does “random” look like?
This is the same idea as the GAMBLER’S FALLACY – i.e. people thinking
that after 10 heads, a tail is more likely to “make-up for all the heads” and
get the average back to 50%. The key is the long run.
Remember the random events that are INDEPENDENT have no memory!
How large is a large number?
• It depends. Particularly on the variability of the outcome.
• The more variable the outcome, the more trials are needed
for the sample mean to approach the actual mean.
HW
• 7.24, 7.27, 7.28, 7.32, 7.34
• Read the article if the idea interests you.
• www.rawbw.com/~deano/articles/aa121896.htm