In: Statistics and Probability

Suppose a red six-sided die, a blue six-sided die, and a yellow six-sided die are rolled. Let

- X1 be the random variable which is 1 if the numbers rolled on the blue die and the yellow die are the same and 0 otherwise;

- X2 be the random variable which is 1 if the numbers rolled on the red die and the yellow die are the same and 0 otherwise;

- X3 be the random variable which is 1 if the numbers rolled on the red die and the blue die are the same and 0 otherwise;

Show that these three variables are pairwise independent but not independent.

One die is rolled and one marble –
yellow, blue, purple, or red is selected at random.
-Determine the number of possible
arrangements.
-Determine the probability of
obtaining the number 2 and the blue marble.
-Determine the probability of
obtaining the number 3 or the red marble.
-Determine the probability of
obtaining an odd number or the purple marble.

A regular six-sided die and a regular eight-sided die are rolled
to find the sum. Determine the probability distribution for the sum
of the two dice. Create a frequency histogram for the probability
distribution and determine the expected sum of the two dice.

Suppose a 6-sided die and a 7-sided die are rolled. What is the
probability of getting sum less than or equal to 5 for the first
time on the 4th roll? Show your work to receive credit.

Three fair, six-sided dice colored red, green and blue are
rolled. Calculate each of the following probabilities: (a) The
probability all three dice show the same face (“triples”). (b) The
probability that the red die shows a larger number than the green
die. (c) The probability that the red die shows a larger number
than the green die and the green die shows a larger number than the
blue die. (d) The probability that the sum of the pips on...

A six sided die is rolled 4 times. The number of 2's rolled is
counted as a success.
Construct a probability distribution for the random
variable.
# of 2's
P(X)
Would this be considered a binomial random variable?
What is the probability that you will roll a
die 4 times and get a 2 only once?
d) Is it unusual to get no 2s when rolling a die 4 times? Why or
why not? Use probabilities to explain.

A six sided die is rolled 4 times. The number of 2's rolled is
counted as a success.
Construct a probability distribution for the random
variable.
# of 2's
P(X)
Would this be considered a binomial random variable?
What is the probability that you will roll a
die 4 times and get a 2 only once?
d Is it unusual to get no 2s when
rolling a die 4 times? Why or why not? Use probabilities to
explain.

Dice Suppose that a red die and a green die are rolled and the
numbers on the sides that face upward are observed. (See Example 7
of this section and Example 2 of the first section.) (a) What is
the probability that the numbers add up to 9? (b) What is the
probability that the sum of the numbers is less than S?

A fair six-sided die is rolled repeatedly until the third time a
6 is rolled. Let X denote the number of rolls required until the
third 6 is rolled. Find the probability that fewer than 5 rolls
will be required to roll a 6 three times.

You roll one blue 6-sided die, and one red 4-sided die. Let A be
the event that the outcome of the red die is twice the outcome on
the blue die Let B be the event that the outcome of the blue die is
greater than the outcome on the red die Let C be the event that the
sum of the two dice is a prime number. a) Find P(B) and P(C) by
clearly listing all outcomes in B...

Example 4: A fair six-sided die is rolled six times. If
the face numbered k is the outcome on roll k for k = 1, 2, 3, 4, 5,
6 we say that a match has occurred. The experiment is called a
success if at least one match occurs during the six trials.
Otherwise, the experiment is called a failure. The outcome space is
O = {success, failure}. Let event A = {success}. Which value has
P(A)?
**This question has...

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