Download Activity for Binomial Distributions

Activity for Binomial Distributions
You have just walked into your AP Government class and the mean teacher is giving you
a pop quiz. It will be a true/false quiz so you end up a little happier than earlier. You
figure that at worst you can get 50% of the answers right since you have a one-half
chance on each question to answer it correctly. There are five questions. If success =
correct answer and failure = incorrect answer, then P(success) = 0.5. We will define the
random variable X as the number of correct answers. Then we want to simulate pop
quizzes with 4 correct answers (since four correct answers will get you a “B”). Our goal
is to determine the long-term relative frequency of a pop quiz in which you get 4
questions correct, that is P(X = 4).
1. You will need to pair up. One person will have the set of 10 pop quizzes with answers
as True or False. The other partner will guess the correct answers. For each test, keep track
of correct responses. After your pop quizzes are over, now reverse roles. When completed
with both partners completing the pop quizzes, tally your results with the other groups.
Calculate the frequency of the event, getting exactly 4 questions correct, P(X = 4).
2. For variety, do the same thing as before, but this time using a calculator. Using the
codes 1 = correct answer and 0 = incorrect answer, enter the command randInt (0, 1, 5).
This command instructs the calculator to randomly pick a whole number from the set {0,
1} and do this 10 times. The outcome {0, 0, 1, 0, 1}, using our codes mean {incorrect,
incorrect, correct, incorrect, correct} in that order. Continue to press ENTER and count
until you have 40 trials. Use a tally mark to record each time you observe exactly 4
correct answers. Calculate the relative frequency for the event {4 correct answers}.
3. Determine the total number of outcomes in this experiment. List the outcomes in the
sample space (One such outcome is CCIIC. For this outcome, X = 3). Then complete the
probability distribution table for the random variable X = number of correct answers.
X
P(X)
0
1
2
3
4
5
Do the results of your simulations come close to the theoretical value for P(X = 4)?
Flipping a coin 5 times and letting heads represent “correct answer” and tails represent
“incorrect answer” would produce exactly the same results. The characterizing features of
this experiment are as follows: (1) a trial consists of flipping the coin once. (2) There are
two outcomes: heads = correct answer (success) and tails = incorrect answer (failure). We
flip the coin 5 times, once for each of the five questions. (3) The coin flips are
independent. And last, (4) the probability of success (correct answer) is the same for each
coin flip (trial). A situation where these four conditions are satisfied is said to be a
binomial setting.
Activity for Geometric Distribution
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.” We want to simulate an
experiment to determine the number of boxes of Cheerios you would expect to buy in
order to get one of the “free” dollar bills.
1. Let a two-digit number, 00 to 99, represent a box of Cheerios. What digits would you
use to represent a box of Cheerios with a $1 bill in it? What digits would you then use to
represent boxes without the $1 bill in it?
2. Starting at the third block of digits in Line 127 of Table B, write down the pairs of
numbers and tally the number of boxes it takes until you get a box with a $1 bill in it.
How many boxes did you have to buy in order to get one with a $1 bill in it? If you do
not usually buy Cheerios, would this promotion induce you to buy a box in hopes of
getting one with a dollar in it?
3. Now using your calculator, using codes 1 to 5 = $1 bill in box and 6 to 100 = $1 bill
not in box, enter the command randInt (1, 100, 1). This command instructs the calculator
to randomly select a whole number from 1 to 100 and do it once. The outcomes 1 to 5 are
a success and 6 to 100 are failures. Continue to press ENTER and count how many boxes
of Cheerios you would need to purchase until you get a box with a $1 in it. Do this for 20
trials. Calculate the mean number of boxes until you get a dollar bill. Get the results from
other groups and tally your answers.
(1) Opening a box to find out if there is a dollar has either a success or failure. (2) The
probability for finding the dollar is the same for each observation (1 out of 20). (3) The
observations are independent of each other (success or failure on one box has no
influence on success or failure on the next box). (4) We are trying to find how many
boxes we need to open before we get a dollar (first success). A situation where these four
conditions are satisfied is said to be a geometric setting.