##### Question

In: Statistics and Probability

# a die is rolled 6 times let X denote the number of 2's that appear on...

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

## Solutions

##### Expert Solution

Solution:

1)

A die is rolled 6 times.

Number of trials = n = 6

let X denote the number of 2's that appear on the die.

For the event "getting 2 " ,

probability of success = p = 1/6

probability of failures = q = 1 - p = 5/6

Trials are independent.

Probability of success is constant from trial to trial.

So , X is binomial with n = 6 , p = 1/6

2)

Using binomial probability formula ,

P(X = x) = (n C x) * px * (1 - p)n - x ; x = 0 ,1 , 2 , ....., n

Now ,

P(at least one 2)

= P(X 1)

= 1 - { P(X < 1) }

= 1 - { P(X = 0) }

= 1 - { (6 C 0) * (1/6)0 * (5/6)6 - 0 }

= 1 - { 0.33489797668 }

= 0.6651

P(at least one 2) =  0.6651

3)

Mean = = n * p = 6 * (1/6) = 1

Standard deviation = = n * p * q = [6 * (1/6) * (5/6)] = 0.9129

Mean = 1

Standard deviation = 0.9129

## Related Solutions

##### A six sided die is rolled 4 times. The number of 2's rolled is counted as...
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 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.
##### 2 dice are rolled. Let X be the number on the first die, Y - on...
2 dice are rolled. Let X be the number on the first die, Y - on the second, and Z=X - Y. Find the expectation and standard deviation of Z.
##### A fair 4-sided die is rolled, let X denote the outcome. After that, if X =...
A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and...
##### Two fair dice are rolled at once. Let x denote the difference in the number of...
Two fair dice are rolled at once. Let x denote the difference in the number of dots that appear on the top faces of the two dice. For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the probability distribution for x. Arrange x in increasing order and write the probabilities P(x) as simplified fractions.
##### A regular six-faced fair die will be rolled 144 times. Let X be the sum of...
A regular six-faced fair die will be rolled 144 times. Let X be the sum of the 144 numbers obtained. Find the approximate probability that X is between 463 and 545.
##### Two dice are rolled. Let the random variable X denote the number that falls uppermost on...
Two dice are rolled. Let the random variable X denote the number that falls uppermost on the first die and let Y denote the number that falls uppermost on the second die. (a) Find the probability distributions of X and Y. x 1 2 3 4 5 6 P(X = x) y 1 2 3 4 5 6 P(Y = y) (b) Find the probability distribution of X + Y. x + y 2 3 4 5 6 7 P(X...
##### Roll three (6-sided) dice. Let X denote the maximum of the values that appear. a. Find...
Roll three (6-sided) dice. Let X denote the maximum of the values that appear. a. Find P(X=1).?? b. Find P(X=2).?? c. Find P(X=3). d. Find P(X=4).?? e. Find P(X=5).? f. Find P(X=6). [Hint: It might be helpful to first find the values of P(X?x).]
##### A die is rolled and, independently, a coin is tossed. Let X be the value of...
A die is rolled and, independently, a coin is tossed. Let X be the value of the die if the coin is H and minus the value of the die if the coin is T. (a) Calculate and plot the PMF of X. (b) Calculate E [X] and var(X) (c) Calculate and plot the PMF of X 2 − 2X. 2. A drunk walks down a street. Assume he starts at block 0. Every 10 minutes, he moves north a...
##### A fair coin is tossed four times. Let X denote the number of heads occurring and...
A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) determine the joint distribution of X and Y (ii) Find Cov(X,Y) and ρ(X,Y).