Illinois State University Mathematics Department
MAT 312: Probability and Statistics for
Middle School Teachers
Spring 1999
9:35  10:50 am TR STV 350A
Dr. Roger Day (day@math.ilstu.edu)

1. TRUE or FALSE: Experimental
probability considers all possible things that could happen
in an experiment.

a) TRUE

b) FALSE

The
description above is of theoretical probability,
for it considers what could happen rather than what
we know has happened.


2. TRUE or FALSE: An experiment consists
of flipping a coin and recording the result. The sample
space for the experiment is the set {heads,
tails}.


3. TRUE or FALSE: Two coins are flipped
and the number of heads showing is recorded, so the sample
space is {0,1,2}. Each outcome in the sample space is
equally likely.

a) TRUE

b) FALSE

Using a table, tree
diagram, or other representation, we can show that
there are four outcomes to this experiment, where H
represents the result head and T represents the
result tail: HH, HT, TH, TT. Each of these four
outcomes is equally likely. For the sample space
{0,1,2}, however, as described in the question, the
outcomes are not equally likely because there are
two ways to get 1 head and just one way to get
either 2 heads or 2 tails.


4. TRUE or FALSE: Two dice are rolled and
the sum of the two faceup sides is recorded. Let event A
represent that the sum is greater than 8. Then P(A) =
4/11.

a) TRUE

b) FALSE

Return to the notes
from Tuesday, 30 March, to look at the sample space
for this experiment. We see that of the 36 possible
outcomes to this experiment, 10 of them meet the
condition stated: the sum is greater than 8.
Therefore, P(A) = 10/36 = 5/18. The incorrect
response of 4/11 was likely determined by assuming
the 11 outcomes are equally likely, and that four
of the 11 outcomes satisfy the required
condition.


5. TRUE or FALSE: Two dice are rolled and
the sum of the two faceup sides is recorded. If event A
represents that the sum is an even number and event B
represents that the sum is an odd number, A and B are
complementary events.

a) TRUE

b) FALSE

Two conditionas are met
here, assuring us that the two events are
complementary. The two events cover all the
outcomes in the sample space, for a sum can only be
odd or even. Second, the two events have nothing in
common, for a particular sum cannot be both odd and
even.


6. TRUE or FALSE: Two dice are rolled and
the sum of the two faceup sides is recorded. If event A
represents that the sum is an even number and event B
represents that the sum is an odd number, A and B are
mutually exclusive events.

a) TRUE

b) FALSE

This is true for the
second reason stated for question (5) above: The
two events have nothing in common, for a particular
sum cannot be both odd and even.


7. If two events are mutually exclusive,
then they are complementary.

a) SOMETIMES
TRUE

b) ALWAYS TRUE

c) NEVER TRUE

From questions (5) and
(6) above, we know that this is at least sometimes
true, for we have one example. To show it is not
always true, we need an example where it is not
true. Try this for the same experiment as used in
questions (5) and (6): Let M be the event that a 3
is rolled and let N be the event that a 4 or 6 is
rolled. Then M and N are mutually exclusive, for
the events have nothing in common. The two events
are not complementary, however, for the two events
do not include all possible outcomes of the
experiment.
Now convince yourself
that the converse of the above statement,
If two events
are complementary, then they are mutually
exclusive, is
ALWAYS TRUE.


8. An experiment consists of two tasks.
First, a fair coin is tossed and the result is recorded.
Next, a die is rolled and the value on the faceup side is
recorded. If A represents the event "a head results" for the
first task and B represents the event "an even number is
recorded" for the second task, then A and B are independent
events.

a) SOMETIMES TRUE

b) ALWAYS
TRUE

c) NEVER TRUE

These are independent
events and always will be because the result of the
coin toss has no influence on the result of the die
roll.


9. For events X and Y, P(X or Y) is
greater than P(X and Y). Assume that X and Y do not
represent the same event and that events X and Y are not
impossible events.

a) SOMETIMES TRUE

b) ALWAYS
TRUE

c) NEVER TRUE

This is
always true because of the inclusive nature of the
OR conjunction and the exclusive nature of the AND
conjunction.
For the event
X OR Y, we must have an outcome in X or in Y, but
not necessarily in both. For the event X AND Y, an
outcome must be in both X and in Y. As long as X
and Y represent different events, it always will be
the case that the OR conjunction includes more
outcomes than the AND conjunction.
Here's an
example for you to work thruogh to help illustrate
this. Suppose a fair die is rolled and the faceup
result is recorded. The sample space is
{1,2,3,4,5,6}. Let event X represent that the
outcome is an odd number and let event Y represent
that the outcome is a prime number. You take it
from here!


10. For events R and T, to calculate
P(RT) we must know P(T).

a) SOMETIMES
TRUE

b) ALWAYS TRUE

c) NEVER TRUE

Some will argue that
this is always true because P(RT) is the
conditional probability of event R given that event
T has occurred. The formula for P(RT) is P(RT) =
P(R and T)/P(T).
Some may argue that
this is never true. Consider the experiment
of rolling a fair die. Suppose we want to determine
the conditional probability that a 3 results, given
that the result is an odd number. We can reason
that the probability is 1/3. By knowing that an odd
number resulted, we can narrow our sample space to
{1,3,5}, each equally likely to occur. Now we ask
the probability of getting a 3 for this new
(restricted) sample space. There is one way out of
the three to get a 3, so the probability is 1/3. I
did not have to know the probability of getting an
odd number to calculate this
probability.
While the second
argument may be compelling, how would you answer
the following question if I hadn't provided
P(T)?
If P(R and T) =
0.25 and P(T) = 0.5, determine
P(RT).


11. A single letter is drawn from the set
{g,e,o}. What is the probability that the letter drawn is a
member of the set {g,e,o,m,t,r,y}?

The desired probability is 1,
because the set {g,e,o} is entirely contained in the set
{g,e,o,m,t,r,y}. No matter what letter is drawn from the
first set, it it a sure thing that the letter will be in the
second set.

12. An experiment consists of rolling a
single fair die. Determine the probability that the result
is:

a) greater than or equal to 5:
2/6 = 1/3

b) an odd number:
3/6 = 1/2

c) a number divisible by 2 or 3:
4/6 = 2/3

d) a number divisible by 2 and 3:
1/6


13. A coin is flipped three times and the
number of heads is recorded.

a) List the sample space for this
experiment:
{0,1,2,3}

b) Determine the probability of each
unique outcome in the sample space.
P(0) = 1/8, P(1) = 3/8, P(2)
= 3/8, P(3) = 1/8


14. An experiment has six mutually
exclusive outcomes, call them A, B, C, D, E, and F. If the
sample space for the experiment is a uniform sample space,
what is P(A or D)?

Beacuse the sample space is
uniform and the events are mutually exclusive, we know all
outcomes are equally likely and each is equal to 1/6. This
means we add the probabilities of the individual outcomes to
determine P(A or D), resulting in a probability of
1/3.

15. Pat is about to take a 4intem
TRUE/FALSE quiz. Pat has no knowledge of the material and
decides to flip a coin to answer each item. What is the
probability that Pat earns a perfect score on the
quiz?

There are 16 outcomes possible
when a coin is flipped four times in succession and the
results recorded:
HHHH

HTHH

THHH

TTHH

HHHT

HTHT

THHT

TTHT

HHTH

HTTH

THTH

TTTH

HHTT

HTTT

THTT

TTTT

If we think of H as TRUE and T
as FALSE, these represent thd 16 ways Pat could have
randomly chosen responses on the quiz. Of course, only one
of these arrangements represents a perfect response from
Pat. The desired probability is therefore
1/16.
