Minds On
A:
Definitions:
SUCCESS: a desired outcome.
B:
FAILURE:
NOT a desired outcome.
Coin Flipping Activity
1. Define a success as flipping a head and a failure as flipping a tail.
2. With a partner, flip a coin ten times and record your results in the table below.
Trial #
1
2
3
4
5
6
7
8
9
10
Outcome
Cumulative #
of Successes
% Success
Cumulative #
of Failures
% Failure
3. Draw a probability tree diagram of the theoretical probabilities for flipping a coin three times.
4. Draw a probability histogram of # of successes vs. probability using the probabilities from the
tree diagram.
Action - GIZMO activity
Exploring Binomial Probability
In this Gizmotm, you will explore probability in a binomial experiment.
1. On the CONTROLS tab, with n = 2, P(S) = 0.50, and P(F) = 0.50, observe the tree diagram to the
right.
a. When n = 2, how many trials does the experiment have?
b. What do P(S) and P(F) represent? Notice that P(S) and P(F) are both 0.50. In your own words,
what does this mean? In what real-life example might this happen?
c. What do SS, SF, FS, and FF represent? Of the total number of possible outcomes, how many
outcomes are two successes? Write your answer as a fraction. Write your answer as a decimal.
2. Vary P(S) by moving the P(S) slider.
a. As P(S) increases, what happens to P(F)? What happens to P(F) as P(S) decreases?
b. When P(S) = 0.23, what is P(F)? (To quickly set a value, type a number in the box to the right of
the slider and press ENTER.) What is P(S) when P(F) = 0.38?
c. Write an equation that describes the numerical relationship between P(S) and P(F). Explain in
your own words why this is true.
d. P(S) and P(F) both have a maximum value of 1 and a minimum value of 0. If the probability of
an event is 1, what does that mean? What does it mean if the probability of an event is 0??
3. Click on the BAR GRAPH tab and observe the bar graph to the right.
a. What do the numbers 0, 1, and 2 along the bottom of the graph represent? What does the
height of each bar represent?
b. If P(S) = 1.00, how would you expect the bar graph to change? What about if P(F) = 1.00? Use
the Gizmo to check your answer.
4. Set n = 3 and P(S) = 0.40. Click on the CALCULATIONS tab and select P(2 Successes).
a. In the calculations at the bottom of the screen, the formula for binomial probability uses the
expression 3C2 as part of the equation. What is the value of 3C2? How many outcomes in the
tree diagram include exactly two out of three successes? How would you describe the meaning
of 3C2 in your own words?
b. What would the equation look like if you wish to find the probability of 1 success? What would
the equation look like if you wish to find P(0 Successes)? Use the Gizmo to check your answer.
c. What would the equation look like if you wish to find the probability of 1 success out of 4 trials,
when P(F) = 0.33? Use the Gizmo to check your answer.
Real-World Binomial Experiments
In this section, you will solve real-life problems involving binomial probability.
1. Suppose that, without even reading the questions, you guess on a five-question true-or-false test.
What is the probability that you will answer all 5 questions correctly? Follow the steps below to find
out.
a. On the CONTROLS tab, set n to represent the number of questions and P(S) and P(F) to
represent the probability of guessing correctly or guessing incorrectly. What is the value of n?
P(S)? P(F)?
b. Click on TREE DIAGRAM and determine the number of possible outcomes. How many of these
outcomes are 5 successes? Write your answer as a fraction. Write your answer as a decimal.
c. Click on the BAR GRAPH tab. Which bar represents answering all 5 questions correctly? How
does the height of that bar compare with your answer to 1b above?
d. Click on the CALCULATIONS tab and select the probability that represents answering all 5
questions correctly. What is P(5)?
e. On paper, write the binomial probability equation that represents answering 4 out of 5 questions
correctly. Evaluate it. Use the Gizmo to check your answers.
2. Suppose that a basketball player shoots 4 free throws. The player's probability of making each
shot is 0.70.
a. What is a trial in this experiment? What is a success? What is a failure?
b. What is n? What is P(S)? What is P(F)?
c. What is the probability that the player makes all 4 free throws? Show the calculation and the
answer. Check your answer using the Gizmo.
d. What is the probability that the player misses all 4 free throws? Show the calculation and the
answer. Check your answer using the Gizmo.
e. How many free throws is the player most likely to make? How can you see this using the bar
graph?
f. Suppose the player wants to improve her free throw shooting so that her probability of making 4
out of 4 free throws is at least 0.5. What is the smallest value of P(S) that will make that true?
Explain how you discover this using the P(S) slider and the bar graph.
Consolidate Debrief - Assessment Questions (5):
1. In a binomial experiment (an experiment with two possible outcomes), if P(S) = 0.36, what is P(F)?
o
A. 0.06
o
B. 0.74
o
C. 0.36
o
D. 0.64
o
2. Which of the following represents the probability of 3 successes in 5 trials?
o
A. 3C5P(S)3P(F)5
o
B. 5C3P(S)3P(F)2
o
C. 3C5P(S)2P(F)5
o
D. 5C3P(S)3P(F)5
o
3. Suppose that, without even reading the questions, you guess on all four questions of a true-or-false test. Compare
the quantity in Column A with the quantity in Column B. Choose the best answer.
o
A. The quantity in Column A is greater.
o
B. The quantity in Column B is greater.
o
C. The two quantities are equal.
o
D. The relationship cannot be determined from the information given.
o
4. Suppose that you ask three friends to go to the mall. Each one has a 0.80 chance of saying yes. What is the
probability that all three friends will say they can go to the mall with you?
o
A. 0.096
o
B. 0.384
o
C. 0.512
o
D. 0.64
You flip a fair coin four times. The coin landing heads up is a called a success. The possible outcomes are shown in
the tree diagram below. What is the probability of the coin landing heads up exactly twice out of the four flips?
Practice
Mike is a basketball star, with a field goal average of 45%, and free throw average of 70%. In any
particular game, if Mike attempts 5 free throws, what is the probability of a certain number of
successful throws?
a) Complete the table and a probability histogram.
x (# of
Probability
successful P(x)
free
throws)
0
5
0
5C 0(0.7) (0.3)  .00243
1
2
3
4
5
Total =
b) Calculate the total area under the probability histogram. How does it compare to the total
in the table?
c) Calculate the expected value using two methods: 𝐸(𝑥) = ∑5𝑖=0 𝑥𝑖 𝑝(𝑥𝑖 ) and 𝐸(𝑥) = 𝑛𝑝.