In: Statistics and Probability

I. Consider the random experiment of rolling a pair of dice. Note: Write ALL probabilities as reduced fractions or whole numbers (no decimals).

1) One possible outcome of this experiment is 5-2 (the first die comes up 5 and the second die comes up 2). Write out the rest of the sample space for this experiment below by completing the pattern:

1-1 |
2-1 |
||||

1-2 |
|||||

1-3 |
|||||

1-4 |
|||||

1-5 |
|||||

1-6 |

2) How many outcomes does the sample space contain? _____________

3) Draw a circle (or shape) around each of the following events (like you would to circle a word in a word search puzzle). Label each event in the sample space with the corresponding letter. Event A has been done for you.

A: Roll a sum of 3.

B: Roll a sum of 7.

C: Roll a sum of at least 10.

D: Roll doubles.

E: Roll snake eyes (two 1’s). F: First die is a 4.

4) Find the following probabilities:

P(A) = _________ P(B) = _________ P(C) = _________

P(D) = _________ P(E) = _________ P(F) = _________

5) The conditional probability of B given A, denoted by P(B|A), is the probability that B will occur when A has already occurred. Use the sample space above (not a special rule) to find the following conditional probabilities:

P(D|C) = _________ P(E|D) = _________ P(D|E) = _________ P(A|B) = _________ P(C|F) = _________

6) Two events are mutually exclusive if they have no outcomes in common, so they cannot both occur at the same time.

Are C and E mutually exclusive? ___________

Find the probability of rolling a sum of at least 10 and snake eyes
on the same roll, using the

sample space (not a special rule).

P(C and E) = __________

Find the probability of rolling a sum of at least 10 or snake eyes, using the sample space. P(C or E) = __________

7) Special case of Addition Rule: If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)

Use this rule to verify your last answer in #6:

P(C or E) = P(C) + P(E) = ________ + ________ = _________

8) Are C and F mutually exclusive? __________ Using sample space, P(C or F) = _________ 9) Find the probability of rolling a “4” on the first die and getting a sum of 10 or more, using the

sample space.

P (C and F) = ________

10) General case of Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) Use this rule to verify your last answer in #8:

P(C or F) = P(C) + P(F) – P(C and F) = ________ + ________ − ________ = _________

11) Two events are independent if the occurrence of one does not influence the probability of the other occurring. In other words, A and B are independent if P(A|B) = P(A) or if P(B|A) = P(B).

Compare P(D|C) to P(D), using the sample space: P(D|C) =
________ . P(D) = ________ .

Are D and C independent? _________

When a gambler rolls at least 10, is she more or less likely to
roll doubles than usual? ___________ Compare P(C|F) to P(C), using
the sample space: P(C|F) = ________ . P(C) = ________ .

Are C and F independent? __________

12) Special case of Multiplication Rule: If A and B are
independent, then P(A and B) = P(A) · P(B).

Use this rule to verify your answer to #9:

P(C and F) = P(C) • P(F) = ________ · ________ = ________ .

13) Find the probability of rolling a sum of at least 10 and getting doubles, using the sample space. P(C and D) = ________ .

14) General case of Multiplication Rule: P(A and B) = P(A) · P(B|A). Use this rule to verify your answer to #13:

P(C and D) = P(C) • P(D|C) = ________ · ________ = ________ .

I. Consider the random experiment of rolling a pair of dice.
Note: Write ALL probabilities as reduced fractions or whole numbers
(no decimals).
1-1
2-1
3-1
4-1
5-1
6-1
1-2
2-2
3-2
4-2
5-2
6-2
1-3
2-3
3-3
4-3
5-3
6-3
1-4
2-4
3-4
4-4
5-4
6-4
1-5
2-5
3-5
4-5
5-5
6-5
1-6
2-6
3-6
4-6
5-6
6-6
2) How many outcomes does the sample space contain?
_____36________
3)Draw a circle (or shape) around each of the following events...

Please answer form 6-14
I. Consider the random experiment of rolling a pair of dice.
Note: Write ALL probabilities as reduced fractions or whole numbers
(no decimals).
1) One possible outcome of this experiment is 5-2 (the first die
comes up 5 and the second die comes up 2). Write out the rest of
the sample space for this experiment below by completing the
pattern:
1-1
2-1
1-2
1-3
1-4
1-5
1-6
2) How many outcomes does the sample space...

probabilities Find the probabilities for each
event. Consider rolling a pair of fair dice two times. Let
A be the total on the up-faces for the first roll and let
B be the total on the up-faces for the second roll.
A = {2 on the first roll}, B = {8 or more on
the first roll}.
A = {2 on the first roll}, B = {8 or more on
the second roll}.
A = {5 or less on the...

Consider a random experiment of rolling 2 dice. What is
probability of rolling a sum larger than 9? Select the best answer.
A. 0.5
B. 0.1667
C. 0.2333
D. None of the above

for the experiment of rolling an ordinary pair of
dice, find the probability that the sum will be even or a multiple
of 6. ( you may want to use a table showing the sum for each of the
36 equally likely outcomes.)

Consider rolling two dice and let (X, Y) be the random variable
pair defined such that X is the sum of the rolls and Y is the
maximum of the rolls.
Find the following:
(1) E[X/Y]
(2) P(X > Y )
(3) P(X = 7)
(4) P(Y ≤ 4)
(5) P(X = 7, Y = 4)

An experiment is rolling two fair dice and adding the spots
together. Find the following probabilities; enter all
answers as simplified fractions using the / bar between numerator
and denominator, with no extra space
Blank #1: Find the probability of getting a sum
of 3.
Blank #2: Find the probability of getting the
first die as a 4.
Blank #3: Find the probability of getting a sum
of 8.
Blank #4: Find the probability of getting a sum
of 3...

Consider rolling a fair dice. You keep rolling the dice until
you see all of the faces (from number 1 to 6) at least once. What
is your expected number of rolls?

Consider the experiment of rolling two dice and the following
events:
A: 'The sum of the dice is 8'
and B: 'The first die is an odd number' and
C: "The difference (absolute value) of the dice is
2"
Find (a) p(A and B) (HINT: You cannot
assume these are independent events.)
(b) p(A or
B)
(c) Are
A and B mutually exclusive events? Explain.
(d) Are A and B independent events?
Explain.
(e) Are B and C independent events?
Explain.

Take a simple experiment of rolling a pair of balanced dice.
Each die has six sides, each side contains one to six spots. Let us
define the random variable x to be the sum of the spots on the two
dice. Display the probability mass function and the distribution
function for the random variable x.

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