Question

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?

Answer #1

**Here' the answer to the question. Let me know in case
you've doubts.**

**We will use the binomial distribution pdf function to
solve the problem**

First lets define probability of success first.

On the 6 numbers of dice, event of success is getting 1 or 2

P(X) = P(getting 1 or 2 out of the 6 numbers on dice) = 2/6 = 1/3

So, p = 1/3

Therefore, probability of failure is 1- P(success) = 1-p = 1-1/3 = 2/3

Hence, out of the 9 rolls P(getting 4 exactly success or exactly 4 failures)

= P(X=4 success) + P(X=4 failures)

= 9C4 * (1/3)^4 *(2/3)^5 + 9C4 * (2/3)^4 *(1/3)^5

= 0.3069

**Answer: P(exactly 4 success or exactly 4 failures on 9
die rolls) = 0.3069**

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.

Example 1 A fair six-sided die is rolled six times. If the face
numbered k is the outcome on roll k for k=1, 2, ..., 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 sample space S={success,
failure} The event A happens when the match happens. A= {success}
Assign a value to P(A)
Simulate the experiment on...

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

Suppose you plan to roll a fair six-sided die two times. What is
the probability of rolling a ‘1’ both times?
Group of answer choices

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.

Suppose we roll a fair six-sided die and sum the values obtained
on each roll, stopping once our sum exceeds 376. Approximate the
probability that at least 100 rolls are needed to get this sum.
Probability =

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.
(a) Find the probability that the outcome 1 is first observed after
5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.
(c) Find the expected number of rolls until the outcome 3 is
observed three times.
(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls
given that it is first observed...

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.
(a) Find the probability that the outcome 1 is first observed after
5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.
(c) Find the expected number of rolls until the outcome 3 is
observed three times.
(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls
given that it is first observed...

Roll a single six-sided die 4 times and record the number of
sixes observed. Does the number of sixes rolled in 4 tosses of a
die meet the conditions required of a binomial random variable?
Construct the probability distribution for this experiment.

A six-sided die is rolled 120 times. Fill in the expected
frequency column. Then, conduct a hypothesis test at the 5% level
to determine if the die is fair. The data below are the result of
the 120 rolls. (Enter exact numbers as integers, fractions, or
decimals.)
Face Value
Frequency
Expected Frequency
1
14
?
2
32
?
3
15
?
4
15
?
5
30
?
6
14
?
Part (a)
State the null hypothesis. Choose 1 or 2...

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