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) = ________ · ________ = ________ .

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 game consisting of rolling a pair of fair 6-sided
dice and recording the sum. It will cost you $1 to play the game.
If the sumis less than 5, then you will win $3. However, you do not
get your $1 back so your profit is $2. If you roll a sum of
exactly8, thenyou will win $4. However, you do not get your $1 back
so your profit is $3. Otherwise, you lose your $1.
A.What is...

Consider the experiment of rolling two dice. You win if you roll
a sum that is at least 7 and at most 12. Find the probability of a
win.

a) A pair of fair dice is thrown. What is the probability of
rolling a value between 8 and 11, inclusive? (Write your answer as
a decimal rounded to 3 decimal places.)
b) What is the probability of drawing a black face card when a
single card is randomly drawn from a standard deck of 52 cards?
(Write your answer as a decimal rounded to 3 decimal places.)

1.) In rolling a pair of dice, what is the probability of a
total of 4 or less?
2.) A blue die and a red die are tossed together. Find the
probability that the sum is less than 7, given that the blue die
shows a 3.
3.) A fair coin is tossed three times. Find the probability of
getting at least two tails, given that the first toss is tails.

An experiment was done to determine if a pair of dice is
“fair”, that is, the probability that each number on each die is
the same. Tom tosses the dice 50 times and determines that the mean
sum of the sample is 6.6, with a standard deviation of 1. 133.
According to the mechanics of tossing two dice, the mean should be
7. Determine if this is a fair dice at the .05 level of
significance. Also determine the p-value....

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