Finite Math B
Probability Distributions
Name: ______________________________________
Trial 1 (Theoretical): Roll two fair dice.
Look at the Cards/Dice handout. Count the number of time each possible sum occurs and verify it in the table below.
Then calculate the probability of each sum occurring.
Make a histogram showing this
PROBABILITY DISTRIBUTION.
TEACHER NOTES:
Make a brief note to students here about rounding and how it
affects accuracy. The true P(2) = 1/36. The value 0.03 is just an
estimate.
Trial 2 (Experimental): Roll two fair dice.
In your group, roll your dice 20 times and calculate the sum. Record your tally:
2: ____________
3: ____________
7: ___________ 8: __________
Possible Sum
Frequency
(Class Totals)
2
3
4
5
6
7
8
9
10
11
12
Total
© Amanda Leahy, 2016
4: ___________ 5: ____________
6: _____________
9: _________
10: ___________
11: ___________
Probability
P(x)
Make a histogram showing this PROBABILITY DISTRIBUTION.
TEACHER NOTES:
Have students use the blanks above to “tally” their results.
When finished, have students record their data somewhere in the
classroom. I like to mark off sections on the front white board or
tape colorful “results sheets” to a classroom wall.
Make each group responsible for finding the classroom sum of
the one or more possible outcomes.
=1
12: ___________
DICE DISCUSSION:
Compare the results of Trial 1 and Trial 2. Are they the same? If they are not the same, how are they similar? How
are they different? How would you describe the shape of your histogram?
Record your observations here:
TEACHER NOTES:
Usually the results are SIMILAR with Trial 1 being perfectly
symmetrical with a line of symmetry at “7”. Trial 2 answers will
vary. Great opportunity to introduce the idea of the “Law of
Large Numbers” and have students make predictions of what
would happen to the histogram if we greatly increased the size of
the sample.
“EXPECTED VALUE:”
The expected value of a probability distribution is a weighted average of the values.
The formula says:
Expected value =  x  P( x)
Multiply each value by its probability and find the sum.
Trial 1: Expected value
Expected value =
2(.03)+3(.06)+4(.08)+5(.11)+6(.14)+7(.17)+8(.14)+9(.11)+10(.08)+11(.06)+12(.03)
=7.07
Why is this not EXACTLY 7?
TEACHER NOTES:
We used rounded probabilities. If we had kept the original fractions,
then our E(x) = 7 exactly. I often will demonstrate this by quickly
plugging it into my calculator and displaying the results.
Trial 2: Calculate the expected value of your probability distribution
Do you think this will be higher or lower than your value for Trial 1?
Expected value =
TEACHER NOTES:
Answers vary based on classroom results.
2( _____) + 3(_____) + 4(_____) + 5(_____) + 6(______) + 7(______) + 8(_____) + 9( ______) + 10(______)
+ 11(_______) + 12(_______)
Expected value = __________
© Amanda Leahy, 2016
TEACHER NOTES:
Trial 3 can be done separately or not at all.
I suggest using “fun size” bags of M&Ms. You want the largest
sample possible, so it is better for each student to have a bag than to
Open your bag of candies and count and record
the
following:
use
larger
bags and have the students work in a group.
Trial 3 (Experimental): How many candies in a bag?
Total number of candies:
_________
CLASSROOM TOTALS:
Caution: Since counting colors is a part of this, I recommend asking
Total number
of RED
candies:
of to
YELLOW
candies:
the previous
class
day if anyone isTotal
colornumber
blind and
ask them
to
come see you. On lab day, have them sit next to someone to help
__________
___________
them
get their counts. After that they should
be good. One year I
had a color blind students say, “They all look brown to me!” and get
unexpectedly frustrated. Now I always ask.
Total Number Frequency Probability
Candies
(How Many)
P(x)
Make a histogram showing this PROBABILITY DISTRIBUTION.
TEACHER NOTES:
Say something about how the bags are packed by weight, so the
number of candies in each bag should be close, but can vary slightly.
Who thinks they have the fewest candy? Keep asking until you have
the smallest. Fill that in the first row. Who has the most? Get your
maximum number. Fill in all the values in the range.
I call out each number one at a time and have students raise their
hand to get how many kids in the class have each value.
Q1) What is the expected number of candies per bag?
(Find the expected value)
Q2) If we randomly choose someone’s bag of candy, what is the probability it will contain LESS than the expected
value?
TEACHER NOTES: There is a possibly of great discussion of the
expected value being in the “middle.” SHOULD 50% of the data be
less than the mean? Etc.
© Amanda Leahy, 2016
CLASSROOM TOTALS:
Total Number Frequency Probability
RED candies (How Many)
P(x)
CLASSROOM TOTALS:
Total Number
Frequency Probability
YELLOW candies (How Many)
P(x)
Q3) What is the expected number of RED candies per bag?
(Find the expected value)
Q4) What is the expected number of BRWON candies per bag?
(Find the expected value)
Q5) If we randomly choose someone’s bag of candy, what is the probability it will contain MORE than the expected
value of RED candy? What is the probability that it will be VERY CLOSE to the expected value of YELLOW candy?
© Amanda Leahy, 2016