Autos arrive at a toll plaza located at the entrance to a bridge at a rate of 50 per minute during the 5:00-to-6:00 P.M. hour. Determine the following probabilities assuming that an auto has just arrived. a. What is the probability that the next auto will arrive within 6 seconds (0.1 minute)? b. What is the probability that the next auto will arrive within 3 seconds (0.05 minute)? c. What are the answers to (a) and (b) if the rate of arrival of autos is 60 per minute? d. What are the answers to (a) and (b) if the rate of arrival of autos is 30 per minute?
Answers
Answer:
a. The probability that the next auto will arrive within 6 seconds (0.1 minute) is 99.33%.
b. The probability that the next auto will arrive within 3 seconds (0.05 minute) is 91.79%.
c. What are the answers to (a) and (b) if the rate of arrival of autos is 60 per minute?
For c(a.), the probability that the next auto will arrive within 6 seconds (0.1 minute) is now 99.75%.
For c(b.), the probability that the next auto will arrive within 3 seconds (0.05 minute) is now 99.75%.
d. What are the answers to (a) and (b) if the rate of arrival of autos is 30 per minute?
For d(a.), the probability that the next auto will arrive within 6 seconds (0.1 minute) is now 95.02%.
For d(b.), the probability that the next auto will arrive within 3 seconds (0.05 minute) is now 77.67%.
Step-by-step explanation:
a. What is the probability that the next auto will arrive within 6 seconds (0.1 minute)?
Assume that x represents the exponential distribution with parameter v = 50,
Given this, we can therefore estimate the probability that the next auto will arrive within 6 seconds (0.1 minute) as follows:
P(x < x) = 1 – e^-(vx)
Where;
v = parameter = rate of autos that arrive per minute = 50
x = Number of minutes of arrival = 0.1 minutes
Therefore, we specifically define the probability and solve as follows:
P(x ≤ 0.1) = 1 – e^-(50 * 0.10)
P(x ≤ 0.1) = 1 – e^-5
P(x ≤ 0.1) = 1 – 0.00673794699908547
P(x ≤ 0.1) = 0.9933, or 99.33%
Therefore, the probability that the next auto will arrive within 6 seconds (0.1 minute) is 99.33%.
b. What is the probability that the next auto will arrive within 3 seconds (0.05 minute)?
Following the same process in part a, x is now equal to 0.05 and the specific probability to solve is as follows:
P(x ≤ 0.05) = 1 – e^-(50 * 0.05)
P(x ≤ 0.05) = 1 – e^-2.50
P(x ≤ 0.05) = 1 – 0.0820849986238988
P(x ≤ 0.05) = 0.9179, or 91.79%
Therefore, the probability that the next auto will arrive within 3 seconds (0.05 minute) is 91.79%.
c. What are the answers to (a) and (b) if the rate of arrival of autos is 60 per minute?
For c(a.) Now we have:
v = parameter = rate of autos that arrive per minute = 60
x = Number of minutes of arrival = 0.1 minutes
Therefore, we specifically define the probability and solve as follows:
P(x ≤ 0.1) = 1 – e^-(60 * 0.10)
P(x ≤ 0.1) = 1 – e^-6
P(x ≤ 0.1) = 1 – 0.00247875217666636
P(x ≤ 0.1) = 0.9975, or 99.75%
Therefore, the probability that the next auto will arrive within 6 seconds (0.1 minute) is now 99.75%.
For c(b.) Now we have:
v = parameter = rate of autos that arrive per minute = 60
x = Number of minutes of arrival = 0.05 minutes
Therefore, we specifically define the probability and solve as follows:
P(x ≤ 0.05) = 1 – e^-(60 * 0.05)
P(x ≤ 0.05) = 1 – e^-3
P(x ≤ 0.05) = 1 – 0.0497870683678639
P(x ≤ 0.05) = 0.950212931632136, or 95.02%
Therefore, the probability that the next auto will arrive within 3 seconds (0.05 minute) is now 99.75%.
d. What are the answers to (a) and (b) if the rate of arrival of autos is 30 per minute?
For d(a.) Now we have:
v = parameter = rate of autos that arrive per minute = 30
x = Number of minutes of arrival = 0.1 minutes
Therefore, we specifically define the probability and solve as follows:
P(x ≤ 0.1) = 1 – e^-(30 * 0.10)
P(x ≤ 0.1) = 1 – e^-3
P(x ≤ 0.1) = 1 – 0.0497870683678639
P(x ≤ 0.1) = 0.950212931632136, or 95.02%
Therefore, the probability that the next auto will arrive within 6 seconds (0.1 minute) is now 95.02%.
For d(b.) Now we have:
v = parameter = rate of autos that arrive per minute = 30
x = Number of minutes of arrival = 0.05 minutes
Therefore, we specifically define the probability and solve as follows:
P(x ≤ 0.05) = 1 – e^-(30 * 0.05)
P(x ≤ 0.05) = 1 – e^-1.50
P(x ≤ 0.05) = 1 – 0.22313016014843
P(x ≤ 0.05) = 0.7767, or 77.67%
Therefore, the probability that the next auto will arrive within 3 seconds (0.05 minute) is now 77.67%.
Final answer:
The probabilities of an auto arriving within a given time frame can be determined using the exponential distribution formula. When the rate of arrival is 50 per minute, the probability of an auto arriving within 6 seconds is approximately 0.9933 and within 3 seconds is approximately 0.9820. These probabilities increase with a higher rate of arrival and decrease with a lower rate of arrival.
Explanation:
To determine the probabilities of an auto arriving within a given time frame, we can use the exponential distribution formula. The exponential distribution is used to model the time until the next event occurs in a Poisson process, which is applicable in this scenario. The formula for the exponential distribution is: P(X <= t) = 1 - e-λt, where λ is the rate of arrival.
- For a rate of arrival of 50 per minute, we can calculate the probability that the next auto will arrive within 6 seconds (0.1 minute) as: P(X <= 0.1) = 1 - e-50*0.1 = 1 - e-5 ≈ 0.9933.
- Similarly, for the next auto to arrive within 3 seconds (0.05 minute), we can use the formula: P(X <= 0.05) = 1 - e-50*0.05 = 1 - e-2.5 ≈ 0.9820.
- When the rate of arrival is 60 per minute, the probabilities will be slightly higher. The probability of the next auto arriving within 6 seconds would be approximately 0.9955, and within 3 seconds would be approximately 0.9838.
- When the rate of arrival is 30 per minute, the probabilities will be slightly lower. The probability of the next auto arriving within 6 seconds would be approximately 0.9866, and within 3 seconds would be approximately 0.9641.
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TEST QUESTION: Jane is making bows for the holidays. One bow requires 23 yard of ribbon. Jane has 13 yards of ribbon. What is the GREATEST number of projects Jane can complete with this ribbon?
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Answer:
If a new product wants to be tested by a company and decides to show 50 samples of this product to 50 selected customers. The company estimates that the probability that the customer buys the product is 0.67, the objective is to determine approximately how many people expect to buy the product.
Let X the random variable of interest "Number of people that will buy a selected product", on this case we now that:
The expected value is given by this formula:
And the standard deviation for the random variable is given by:
So then they can conclude that for each group of 50 people they expect that about 33-34 peoploe will buy the product with a standard deviation of 3.32.
Step-by-step explanation:
Previous concepts
A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
The probability mass function for the Binomial distribution is given as:
Where (nCx) means combinatory and it's given by this formula:
Solution to the problem
If a new product wants to be tested by a company and decides to show 50 samples of this product to 50 selected customers. The company estimates that the probability that the customer buys the product is 0.67, the objective is to determine approximately how many people expect to buy the product.
Let X the random variable of interest "Number of people that will buy a selected product", on this case we now that:
The expected value is given by this formula:
And the standard deviation for the random variable is given by:
So then they can conclude that for each group of 50 people they expect that about 33-34 peoploe will buy the product with a standard deviation of 3.32.
The first transformation was a
.
The second transformation was a
Answers
The first transformation was a rotation about point A.
The second transformation was a reflection across line M.
What is a rotation?
In Mathematics, a rotation can be defined as a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.
By critically observing the diagram which illustrates the sequence of transformations, we can logically deduce that the first transformation was a clockwise rotation about point A by 180 degrees.
Furthermore, the second transformation that maps W'X'Y'Z' to W''X''Y''Z'' is a reflection across the line of reflection M.
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Find the amount of the time.
Answers
Answer:
The amount of time is 8 years
Step-by-step explanation:
Simple interest refers to the interest produced by initial capital over a period of time. This does not accumulate to the capital to produce the interests of the following period, so the interest generated or paid in all periods is the same, while the interest rate and term dont vary.
The amount of interest paid or charged depends on three important amounts: Capital, rate and time. This is expressed by the following equation:
I = P * r * t
Where:
- I = interest.
- P = initial capital.
- r = interest rate in decimals (it is usually a given data in percentage, which you must divide by 100 to get the value in decimals).
- t = time in years.
In this case I has a value of $ 720, P has a value of $ 1000 and r of 9%, which making the conversion to decimal is 0.09
Replacing in the equation you get:
720=1000*0.09*t
Multiplying 1000 * 0.09 you get:
720=90*t
Dividing both sides by 90, in order to isolate t, you get:
8=t
Remembering that the value of t is expressed in years, this means that the amount of time is 8 years.
I / PR = T
I = 720
P = 1000
R = 9% = 0.09
now we sub
720 / (1000)(0.09) = T
720 / 90 = T
8 = T <===
Answers
Answer:
$6800
Step-by-step explanation:
1. Assuming your question was how much is her cost the total cost should amount to $6800.
2. First you should multiply 10 by 600 to find out the cost for the credit hours
3. Next you take the cost of textbooks into factor, with each being 400 a semester you add them together to get $800
4. You add both costs together and get the final price of $6800 for both semesters of her education.
Final answer:
Linda's total cost for college for this academic year is calculated as $12,800, encompassing both tuition and textbooks. This is part of an observed trend in increasing higher education costs.
Explanation:
Cost of Tuition: Linda is paying $600 per credit hour for 20 credit hours (10 each semester), so $600 * 20 = $12,000 in total for tuition.
Cost of Textbooks: She is also spending $400 per semester for textbooks, so $400 * 2 = $800 in total for textbooks.
Total Cost: Thus, Linda's total cost for the academic year would be the sum of these two costs, i.e, $12,000 + $800 = $12,800. The costs of tuition, textbooks and other expenses are part of the rising trend of higher education costs. Despite this, the value of education still remains high, as it can lead to better job prospects and higher earning potential in the future.
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Answers
Answer:
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction.
Step-by-step explanation:
the group to lead the event?
Answers
Answer:
Step-by-step explanation:
If you are always included to lead, that leaves 10 students to choose from.
If all that is important is being selected, then there are 10C3 = 120 ways to choose them.
If each selection has a unique duty, then there are 10P3 = 720 ways to assign them their jobs.
Answer: 10P3=720
Step-by-step explanation: