Show me the money!!!!
In 1986-1987, Cheerios
cereal boxes displayed a
dollar bill on the front of
the box and a cartoon
character who said, “Free
$1 bill in every 20th box.”
1. What kind of distribution is this?
2. Use your calculator to create a simulation to determine the number of boxes
of Cheerios you would expect to buy in order to get one of the “free” dollar
bills.
3. Run the simulation 5 times and write your results for how many times it
takes to get a $1 bill on the board.
4. Find the Mean and Standard Deviation for this distribution. (Use the
formulas)
The Birth Day Game
Mr. Latham is planning to give you 10
problems for homework. As an alternative,
you can agree to play the Birth Day Game.
Here’s how it works. A student will be
selected at random from your class and
asked to guess the day of the week (for instance, Thursday) on which one of Mr. Latham’s
friends was born. If the student guesses correctly, then the class will have only one
homework problem. If the student guesses the wrong day of the week, then Mr. Latham
chooses another friend’s birthday and another student to guess. If this student gets it
right, the class will have two homework problems.
The game continues until a student correctly guesses the day
on which one of Mr. Latham’s many friends was born. Mr.
Latham will assign a number of homework problems that is
equal to the total number of guesses made by members of your
class.
Are you ready to play the Birth Day Game?
1.
2.
3.
4.
5.
What type of distribution is this?
What is the probability of a student guessing the right day?
What is the probability of a student guessing on the second try P(X=2)?
What is the probability of a student guessing on the third try P(X=3)?
Play the Birth Day game with a partner. One of you pretend to be Mr.
Latham and one pretend to be a student. Play the game 3 times and keep
track of how many times the class will have less than 10 homework
problems. Switch roles and play 3 more times. Post Results on the Board.
A binomial distribution will be approximately correct as a model for one of
these two sports settings and not for the other. Use BINS to determine which
is a binomial distribution.
a.
A National Football League kicker
has made 80% of his field goal attempts
in the past. This season he attempts 20
field goals. The attempts differ widely in
a distance, angle, wind, and so on.
b. A National Basketball Association player has made 80% of his field
goal attempts in the past. This season he takes 20 free throws.
Basketball free throws are always attempted from 15 feet away with
no interference from other players.
1. Find 𝑃(𝑋 = 4) and explain what the probability means in context.
2. Find 𝑃(𝑋 < 4) and explain what the probability means in context.
3. Calculate the mean and standard deviation for the distribution.
Using technology for binomial distribution:
pdf: probability distribution function – assigns a probability to each
value of X. (See chapter 7 notes for requirements.)
Ti-83/84
2nd DISTR then pick binompdf
Format for parenthesis is 𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓(𝑛, 𝑝, 𝑋)
**This calculates binomial distribution for any 𝑃(𝑋 = 𝑘)
cdf: cumulative distribution function – calculates the sum of the
probabilities for 0, 1, 2, …., up to the value X.
Ti-83/84
2nd DISTR then pick binomcdf
Format for parenthesis is 𝑏𝑖𝑛𝑜𝑚𝑐𝑑𝑓(𝑛, 𝑝, 𝑋)
**This calculates binomial distribution for any 𝑃(𝑋 ≤ 𝑘)
Using technology for geometric distribution:
Same as binomial except –
𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓 (𝑝, 𝑋 ) 𝑓𝑜𝑟 𝑃(𝑋 = 𝑛)
𝑔𝑒𝑜𝑚𝑒𝑡𝑐𝑑𝑓 (𝑝, 𝑋 ) 𝑓𝑜𝑟 𝑃(𝑋 ≤ 𝑛)
The probability that it takes more than 𝑛 trials to see the first success
in a geometric setting is:
𝑷(𝑿 > 𝒏) = (𝟏 − 𝒑)𝒏
This is the same as 1 − 𝑔𝑒𝑜𝑚𝑒𝑡𝑐𝑑𝑓(𝑝, 𝑋)
Normal Approximation to Binomial Distributions
*As the number of trials, n, gets larger then the binomial
distribution will get closer to a Normal distribution
𝑵 (𝒏𝒑, √𝒏𝒑(𝟏 − 𝒑))
-Example 12 on pg. 527-528 demonstrates this
*Rule of thumb…. use Normal distribution when
𝑛𝑝 ≥ 10 𝐚𝐧𝐝 𝑛(1 − 𝑝) ≥ 10