Question 1

If the probability of a success on a single trial is .2,what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?

Accepted Answer

The answer of If the probability of a success on...

Question 2

A test has 6 multiple choice questions,each with 4 alternatives.What is the probability of guessing 5 or more questions correctly?

Accepted Answer

The probability of guessing a question correctly is 1/4. Using the binomial probability formula, calculate the probability of getting exactly 5 and exactly 6 questions correct, then sum these probabilities. The probability of getting 5 correct is 6C5 * (1/4)^5 * (3/4)^1 and for 6 correct is 6C6 * (1/4)^6. Adding these gives approximately 0.0046.

Question 3

If the probability distribution of X is $\begin{array} { c c } 
\mathrm { X } & \mathrm { P } ( \mathrm { X } ) \
3 & 1 / 8 \
4 & 1 / 8 \
5 & 3 / 8 \
6 & 3 / 8
\end{array}$
What is the expected value of X?

Accepted Answer

Expected value is calculated as follows:

E(X) = Σ[X * P(X)]

E(X) = (3 * 1/8) + (4 * 1/8) + (5 * 3/8) + (6 * 3/8)

E(X) = 3/8 + 4/8 + 15/8 + 18/8

E(X) = 40/8

E(X) = 5

Therefore, the expected value of X is 5, which is option B.

Question 4

Three candidates run for different offices in different counties.Each has a one in three chance of being elected in his/her county.What is the probability that at least one of them will be elected?

Accepted Answer

The probability that none of them will be elected in any county is (2/3)^3 = 8/27. 
Therefore, the probability that at least one of them will be elected is 1 - 8/27 = 19/27 ≈ 0.7037, which is option B.

Question 5

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 100 calculators are selected at random,what is the standard deviation of the number of defectives?

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula: √(n * p * (1-p)), where n is the number of trials (100), and p is the probability of success (0.10). Thus, √(100 * 0.10 * 0.90) = √9 = 3.00.

Question 6

Determine the probability that a 3 will appear twice,if a fair die is rolled 10 times.

Accepted Answer

The probability of getting a 3 on a single roll of a fair die is 1/6. The probability of getting a 3 twice in 10 rolls is (1/6)^2 * 10C2 = 2907/10000.

Question 7

The number of calls coming into a PBX follows a Poisson process with a mean of 120 calls per hour.What is the probability of no calls in a one-minute interval?

Accepted Answer

We know that the mean number of calls per hour, λ = 120. To find the probability of no calls in a one-minute interval, we need to convert the rate to calls per minute.

λ = 120/60 = 2 calls per minute

The probability of no calls in a one-minute interval is given by the Poisson distribution formula:

P(X = 0) = e^(-λ) * λ^0 / 0!

Substituting the value of λ, we get:

P(X = 0) = e^(-2) * 2^0 / 0! = e^(-2) ≈ 0.1353

Therefore, the correct choice is B, with a probability of approximately 0.1353.

Question 8

Historical data show that the average number of patient arrivals at the intensive care unit of General Hospital is 3 patients every 2 hours.Assume that the patient arrivals are distributed according to Poisson distribution.Determine the probability of at least 4 but no more than 8 patients arriving in a three-hour period.

Accepted Answer

The average rate of arrival is 4.5 patients in 3 hours (since 3 patients in 2 hours translates to 4.5 in 3 hours). Using the Poisson distribution formula, calculate the probabilities for 4, 5, 6, 7, and 8 arrivals and sum them to find the probability of at least 4 but no more than 8 arrivals.

Question 9

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next minute,three cars will arrive?

Accepted Answer

The given problem follows Poisson distribution with the following parameters:
λ = 0.5 (average rate of arrival per minute)

The probability of three cars arriving in the next minute can be calculated using the Poisson distribution formula:
P(X = 3) = (e^(-λ)* λ^X) / X!
P(X=3) = (e^(-0.5)* 0.5^3) / 3!
        = (0.08208499862) / 6
        = 0.0136808331
        ≈ 0.0126 (rounded to four decimal places)

Therefore, the probability that during the next minute, three cars will arrive is 0.0126 or 1.26% (approximately).
Hence, the best choice is (C).

Question 10

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than 6?

Accepted Answer

The answer of If x is a Poisson random variable...

Question 11

Assume the number of trucks passing an intersection has a Poisson distribution with mean of 5 trucks per minute.What is the probability of 0 or 1 trucks in one minute?

Accepted Answer

The probability of 0 or 1 trucks in one minute is the sum of the probabilities of 0 and 1 trucks. The probability of 0 trucks is e^-5 = 0.0067, and the probability of 1 truck is 5e^-5 = 0.0337. Therefore, the probability of 0 or 1 trucks is 0.0067 + 0.0337 = 0.0404.

Question 12

A vaccine is 95 percent effective.What is the probability that it is not effective for one and only one individual out of 20 individuals?

Accepted Answer

The answer of A vaccine is 95 percent effective.What is...

Question 13

If the probability distribution of X is $\begin{array} { c c } 
\mathrm { X } & \mathrm { P } ( \mathrm { X } ) \
3 & 1 / 8 \
4 & 1 / 8 \
5 & 3 / 8 \
6 & 3 / 8
\end{array}$
What is the variance of X?

Accepted Answer

The answer of If the probability distribution of X is...

Question 14

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next five minutes,three cars will arrive?

Accepted Answer

The probability of 3 cars arriving in 5 minutes is given by the Poisson distribution:P(X = 3) = (e^-λ * λ^x) / x!where λ is the average number of arrivals per minute, x is the number of arrivals, and e is the base of the natural logarithm.In this case, λ = 0.5 cars per minute and x = 3 cars.P(X = 3) = (e^-0.5 * 0.5^3) / 3!= (0.6065 * 0.125) / 6= 0.0126

Question 15

For a binomial process,the probability of success is 40 percent and the number of trials is 5.Find the expected value.

Accepted Answer

The expected value of a binomial process is equal to the product of the number of trials and the probability of success. 
Expected value = number of trials x probability of success 
Expected value = 5 x 0.4 
Expected value = 2.0

Question 16

Twenty coins are tossed.What is the probability of getting exactly 10 heads?

Accepted Answer

The answer of Twenty coins are tossed.What is the probability...

Question 17

If x is a Poisson random variable with a mean of 10,what is the probability that x is equal to 8?

Accepted Answer

The probability of a Poisson random variable with a mean of 10 being equal to 8 is given by the Poisson distribution formula:P(X = k) = (e^-λ * λ^k) / k!where λ is the mean of the distribution.In this case, λ = 10, so:P(X = 8) = (e^-10 * 10^8) / 8! = 0.1126Therefore, the probability that x is equal to 8 is 0.1126, which is choice A.

Question 18

Historical data show that the average number of patient arrivals at the intensive care unit of General Hospital is 3 patients every two hours.Assume that the patient arrivals are distributed according to Poisson distribution.Determine the probability of 6 patients arriving in a five-hour period.

Accepted Answer

The probability of 6 patients arriving in a five-hour period can be calculated using the Poisson distribution formula:P(X = k) = (e^-λ * λ^k) / k!where:λ is the average number of arrivals per unit timek is the number of arrivalsIn this case, λ = 3 patients every two hours, and we want to find P(X = 6) for a five-hour period.First, we need to convert the rate from patients every two hours to patients per hour:λ = 3 patients / 2 hours = 1.5 patients / hourNow we can plug in the values into the Poisson distribution formula:P(X = 6) = (e^-1.5 * 1.5^6) / 6!= (0.2231 * 11.3906) / 720= 0.246Therefore, the probability of 6 patients arriving in a five-hour period is 0.246, or approximately 24.6%.

Question 19

A vaccine is 95 percent effective.What is the probability that it is not effective for more than one out of 20 individuals?

Accepted Answer

The answer of A vaccine is 95 percent effective.What is...

Question 20

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than or equal to 2?

Accepted Answer

Using the Poisson distribution formula, we have:

P(X >= 2) = 1 - P(X < 2) = 1 - P(X = 0) - P(X = 1)

where P(X = k) = (e^-λ * λ^k) / k!

λ is the mean of the Poisson distribution, so in this case λ = 10.

P(X = 0) = (e^-10 * 10^0) / 0! = e^-10 = 0.0000453999

P(X = 1) = (e^-10 * 10^1) / 1! = e^-10 * 10 = 0.000453999

Therefore,

P(X >= 2) = 1 - 0.0000453999 - 0.000453999 = 0.9995

So the best choice is C.

Quiz 5: Discrete Random Variables

If the probability of a success on a single trial is .2,what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?

A test has 6 multiple choice questions,each with 4 alternatives.What is the probability of guessing 5 or more questions correctly?

If the probability distribution of X is $\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}$ What is the expected value of X?

Three candidates run for different offices in different counties.Each has a one in three chance of being elected in his/her county.What is the probability that at least one of them will be elected?

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 100 calculators are selected at random,what is the standard deviation of the number of defectives?

Determine the probability that a 3 will appear twice,if a fair die is rolled 10 times.

The number of calls coming into a PBX follows a Poisson process with a mean of 120 calls per hour.What is the probability of no calls in a one-minute interval?

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next minute,three cars will arrive?

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than 6?

Assume the number of trucks passing an intersection has a Poisson distribution with mean of 5 trucks per minute.What is the probability of 0 or 1 trucks in one minute?

A vaccine is 95 percent effective.What is the probability that it is not effective for one and only one individual out of 20 individuals?

If the probability distribution of X is $\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}$ What is the variance of X?

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next five minutes,three cars will arrive?

For a binomial process,the probability of success is 40 percent and the number of trials is 5.Find the expected value.

Twenty coins are tossed.What is the probability of getting exactly 10 heads?

If x is a Poisson random variable with a mean of 10,what is the probability that x is equal to 8?

Historical data show that the average number of patient arrivals at the intensive care unit of General Hospital is 3 patients every two hours.Assume that the patient arrivals are distributed according to Poisson distribution.Determine the probability of 6 patients arriving in a five-hour period.

A vaccine is 95 percent effective.What is the probability that it is not effective for more than one out of 20 individuals?

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than or equal to 2?

Quiz 5: Discrete Random Variables

If the probability of a success on a single trial is .2,what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?

A test has 6 multiple choice questions,each with 4 alternatives.What is the probability of guessing 5 or more questions correctly?

If the probability distribution of X is XP(X) 31/841/853/863/8\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}X3456​P(X) 1/81/83/83/8​ What is the expected value of X?

Three candidates run for different offices in different counties.Each has a one in three chance of being elected in his/her county.What is the probability that at least one of them will be elected?

The J.O.Supplies Company buys calculators from a Korean supplier.The probability of a defective calculator is 10 percent.If 100 calculators are selected at random,what is the standard deviation of the number of defectives?

Determine the probability that a 3 will appear twice,if a fair die is rolled 10 times.

The number of calls coming into a PBX follows a Poisson process with a mean of 120 calls per hour.What is the probability of no calls in a one-minute interval?

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next minute,three cars will arrive?

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than 6?

Assume the number of trucks passing an intersection has a Poisson distribution with mean of 5 trucks per minute.What is the probability of 0 or 1 trucks in one minute?

A vaccine is 95 percent effective.What is the probability that it is not effective for one and only one individual out of 20 individuals?

If the probability distribution of X is XP(X) 31/841/853/863/8\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}X3456​P(X) 1/81/83/83/8​ What is the variance of X?

During off hours,cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute.The arrivals are distributed according to the Poisson distribution.What is the probability that during the next five minutes,three cars will arrive?

For a binomial process,the probability of success is 40 percent and the number of trials is 5.Find the expected value.

Twenty coins are tossed.What is the probability of getting exactly 10 heads?

If x is a Poisson random variable with a mean of 10,what is the probability that x is equal to 8?

Historical data show that the average number of patient arrivals at the intensive care unit of General Hospital is 3 patients every two hours.Assume that the patient arrivals are distributed according to Poisson distribution.Determine the probability of 6 patients arriving in a five-hour period.

A vaccine is 95 percent effective.What is the probability that it is not effective for more than one out of 20 individuals?

If x is a Poisson random variable with a mean of 10,what is the probability that x is greater than or equal to 2?

If the probability distribution of X is $\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}$ What is the expected value of X?

If the probability distribution of X is $\begin{array} { c c } \mathrm { X } & \mathrm { P } ( \mathrm { X } ) \\3 & 1 / 8 \\4 & 1 / 8 \\5 & 3 / 8 \\6 & 3 / 8\end{array}$ What is the variance of X?