Question

A fair 4-sided die is rolled 7 times. |

(a) | Find the probability that the side 1 comes up exactly 3 times. |

(b) | Find the probability that there is at least one side that comes up exactly 3 times. |

Answer #1

Binomial distribution: P(X) = nCx p^{x}
q^{n-x}

a) P(side 1), p = 1/4

q = 1 - 1/4 = 3/4

Number of rolls, n = 7

a) P(the side 1 comes up exactly 3 times) = 7C3 x
(1/4)^{3} x (3/4)^{4}

= **0.1038**

b) P(at least one side that comes up exactly 3 times) = P(1 comes up exactly 3 times) + P(2 comes up exactly 3 times) + P(3 comes up exactly 3 times) + P(4 comes up exactly 3 times) - Number of ways to select 2 numbers from 4 x P(2 numbers comes up exactly 3 times)

= 0.1038x4 - 4C2 x (7C3 x 4C3) x (1/4)^{3} x
(1/4)^{3} x (2/4)

= 0.4152 - 6x0.0171

= **0.3127**

A fair 4-sided die and a fair 6-sided die will be rolled. A)
Give the sample space. B) What is the probability the sum is 9? C)
What are the chances that the sum is 4, or the 4 sided die comes up
a 3?

A six-sided fair die is rolled six times independently. If side
i is observed on the ith roll, it is called a match on the ith
trial, i = 1, 2, 3, 4, 5, 6. Find the probabilities that
(a) all six trials result in matches,
(b) at least one match occurs in these six trials,
(c) exactly two matches occur in these six trials.

1) A fair die is rolled 10 times. Find an expression for the
probability that at least 3 rolls of the die end up with 5 dots on
top.
2) What is the expected number of dots that show on the top of
two fair dice when they are rolled?

4 fair 10-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
3 given that all 4 dice land on different numbers.
(b)
True or False: If X is the sum of the 4 numbers from
one roll, and Y is the maximum of the 4 numbers from one
roll, then X and Y are independent random
variables.

Assume that a fair
six-sided die is rolled 9 times, and the roll is called a success
if the result is in {1,2}{1,2}.
What is the probability that there are exactly 4 successes or
exactly 4 failures in the 9 rolls?

A fair 10-sided die is rolled 122 times. Consider the event
A = {the face 6 comes up at most 2 times}.
(a)
Find the normal approximation for P(A)
without the continuity correction.
(b)
Find the normal approximation for P(A)
with the continuity correction.
(c)
Find the Poisson approximation for P(A).

5 fair 8-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
3 given that all 5 dice land on different numbers.

A fair six-sided die has two sides painted red, 3 sides painted
blue and one side painted yellow.
The die is rolled and the color of the
top side is recorded.
List all possible outcomes of this random experiment
Are the outcomes equally likely? Explain
Make a probability distribution table for the random variable
X: color of the top side
2. If a pair of dice painted the same way as in problem 1 is
rolled, find the probability...

If a single six-sided die is rolled five times, what is the
probability that a six is thrown exactly three times?
a)
0.125
b)
0.032
c)
0.042
d)
0.5

A fair die is rolled 500 times. Find the probability that a
number 2 or less comes up on at most 150 rolls. b. The scores on a
psychological proﬁle test given to job applicants at a nuclear
facility are known to be normally distributed with a mean of 65 and
a standard deviation of 10. 1. What score is required for an
applicant to be in the top 10%? 2. Suppose a random sample of 16
applicants is selected,...

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