In: Statistics and Probability

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) P(2) = 1/6

# of 2's Probability

0
(1 - 1/6)^{4} = 0.4823

1
4C1 * (1/6) * (5/6)^{3} = 0.3858

2
4C2 * (1/6)^{2} * (5/6)^{2} = 0.1157

3
4C3 * (1/6)^{3} * (5/6)^{1} = 0.0154

4
(1/6)^{4} = 0.0008

b) Yes this is a binomial distribution

c) P(X = 1) = 0.3858

d) P(X = 0) = 0.4823

Since probability is not less than 0.05 or greater than 0.95, this is not an unusual event. So it is not unusual to get no 2s when rolling a die 4 times

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.

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...

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...

a die is rolled 6 times let X denote the number of 2's that
appear on the die.
1. show that X is binomial.
2. what is the porbaility of getting at least one 2.
3. find the mean and the standard deviaion of X

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.

A fair six-sided green die is rolled resulting in a number
between 1 and 6. Then that number of red dice are rolled. Find the
expected value and variance of the sum of the red dice.

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.

Suppose a die is rolled six times and you need to find
a) The probability that at least two 4 come up
b) The probability that at least five 4's come up
Solve using the Binomial probability formula.

Suppose a 9-sided and a 4-sided die are rolled. Find these
probabilities. a. P(roll sum of less than 5 or roll doubles)
b. P(roll a sum greater than 6 for the first time on the sixth
roll)

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.

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