4. A coil of wire has 600 metres of wire. Suppose there are 20 nicks (the most common problem with wire) are randomly distributed on a coil.
a) What is the probability that in a 60 metre length of wire there will be at least 3 nicks?
b) What is the probability that in a 47 metre length of wire there will be exactly 4 nick(s)?
5. Two students have started a business to seal driveways during the summer months. They rent a pickup truck and a power sprayer. With this they will use a tar based spray to seal asphalt driveways. Past experience has shown that the best time to sign up customers is to ring their doorbells between 5:00 and 8:00 p.m. on any weekday evening. Any jobs that they obtain will be completed the next day. In the months of June. July and August they find that they get an average of 2.5 customers per hour ringing doorbells.
a) What is the probability that they will get from 4 to 10 jobs in an evening of soliciting?
b) They charge $25 per driveway. If the truck costs $65 per day, and the spraying equipment costs $20 per day and the material to seal one driveway costs $6, what is the probability that they will make a profit on any given day.
a) What is the probability that in a 60 metre length of wire there will be at least 3 nicks?
Well, let's do some math and unravel this coil of wire together.
First, we need to determine the probability of having 0, 1, or 2 nicks in the 60 meter length.
Assuming the nicks are randomly distributed, the probability of a nick occurring in any given meter is 20/600 = 1/30.
To find the probability of no nicks in 60 meters, we can use the binomial distribution formula:
P(X=k) = (n choose k) * (p^k) * ((1-p)^(n-k))
where n is the number of trials (60 meters), k is the number of successes (no nicks), and p is the probability of success (1/30).
Using this formula, we find:
P(no nicks) = (60 choose 0) * ((1/30)^0) * ((1-(1/30))^(60-0))
= (60 choose 0) * (1) * ((29/30)^60)
≈ (1) * (1) * (0.1750)
= 0.1750
Similarly, you can calculate the probability of 1 nick and 2 nicks.
Now, to find the probability of at least 3 nicks, we subtract the sum of these probabilities from 1:
P(at least 3 nicks) = 1 - (P(no nicks) + P(1 nick) + P(2 nicks))
I'll leave the calculations up to your mathematical prowess, but I'm here to provide some comedic relief while you work your way through the numbers. Good luck, math warrior!
b) What is the probability that in a 47 metre length of wire there will be exactly 4 nicks?
Ah, the quest for the magical number of 4 nicks in a 47 meter stretch of wire. Let's see if we can conjure up that probability for you.
Using the same probability of a nick occurring in any given meter (1/30), we can employ the binomial distribution formula once again. This time, we want to find the probability of exactly 4 nicks, so k = 4.
P(X=4) = (n choose k) * (p^k) * ((1-p)^(n-k))
Plugging in our values, we get:
P(exactly 4 nicks) = (47 choose 4) * ((1/30)^4) * ((1-(1/30))^(47-4))
Now it's your turn to work your magic with the numbers. Remember to calculate the combination part (n choose k), and watch out for those exponents. Math can be a tricky circus, but I believe in your ability to tame those calculations!
5. Two students have started a business to seal driveways during the summer months.
Ah, the joys of being a pavement Picasso! Let's see what the probabilities have in store for these two entrepreneurial students.
a) What is the probability that they will get from 4 to 10 jobs in an evening of soliciting?
Well, let's dive into the probability pool and see what we can fish out for our ambitious duo.
If they get an average of 2.5 customers per hour, we can assume that the number of customers follows a Poisson distribution.
The formula for the Poisson distribution probability mass function is:
P(X=k) = (e^(-λ) * λ^k) / k!
Where λ is the average number of customers (2.5 in this case) and k is the number of customers we're interested in (4 to 10).
To find the probability of getting from 4 to 10 jobs, we need to sum up the individual probabilities for each k value:
P(4 to 10 jobs) = P(X=4) + P(X=5) + ... + P(X=10)
Now it's time for some calculation magic! Remember that exclamation mark means you have to perform a factorial operation. Don't forget to pack your math wand and let's cast some equations spells!
b) They charge $25 per driveway. If the truck costs $65 per day, and the spraying equipment costs $20 per day, and the material to seal one driveway costs $6, what is the probability that they will make a profit on any given day?
Ah, the thirst for profits! Let's calculate the probability of quenching that thirst with some cold, hard cash.
To determine if they make a profit, we need to subtract their expenses from their revenue and see if the number that pops out is positive.
Revenue = Number of driveways * Price per driveway
Expenses = Cost of truck + Cost of spraying equipment + (Number of driveways * Cost of material per driveway)
If (Revenue - Expenses) is positive, they make a profit; otherwise, they end up wearing a frown upside down.
Now, to find the probability of making a profit, we need to calculate the ratio of the number of profitable days to the total number of possible days.
Let's crack open that probability piggy bank and see if we have enough cash to make ends meet. Remember, even if the calculations don't yield a profit, laughter is always a worthwhile currency!
To calculate the probabilities in these scenarios, we need to use the concepts of probability and combinations.
For question 4a:
Step 1: Calculate the probability of getting 3 nicks in a 60-meter length of wire.
- Assume the nicks are randomly distributed, so the probability of a nick occurring at any specific point is the same across the entire wire length.
- Let's define success as getting a nick within the 60-meter length. The probability of success for a single point is 3/600 = 1/200.
Step 2: Use the binomial distribution formula to calculate the probability of at least 3 successes.
- Let X be the number of nicks in a 60-meter length of wire.
- P(X ≥ 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2)
- P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
= C(20, k) * (1/200)^k * (1 - 1/200)^(20-k)
By calculating P(X = 0), P(X = 1), and P(X = 2), and subtracting them from 1, you will get the final probability of at least 3 nicks in a 60-meter length of wire.
For question 4b:
To find the probability of exactly 4 nicks in a 47-meter length of wire, follow a similar process as in question 4a, while using the given length of wire and number of nicks.
For question 5a:
Step 1: Calculate the average number of jobs per hour by dividing the total number of jobs (from 4 to 10) by the time interval (3 hours).
- Average number of jobs per hour = (4+5+6+7+8+9+10) / 3
Step 2: Use the Poisson distribution formula to calculate the probability of obtaining a specific number of jobs in an evening of soliciting.
- P(X = k) = (e^(-λ) * λ^k) / k!
- λ is the average number of jobs per hour multiplied by the time interval (3 hours).
- k is the specific number of jobs you want to calculate the probability for.
- e is Euler's number, which is approximately 2.71828.
By plugging in the values into the Poisson distribution formula for each value of k (from 4 to 10) and adding them up, you will get the final probability of getting from 4 to 10 jobs in an evening of soliciting.
For question 5b:
Step 1: Calculate the revenue (R) per job, which is the amount charged per driveway ($25).
Step 2: Calculate the total cost (C) per day, which is the truck cost ($65), spraying equipment cost ($20), and material cost per job ($6) multiplied by the total number of jobs expected.
Step 3: Calculate the profit (P) per day, which is the revenue minus the total cost (P = R - C).
Step 4: Find the probability of making a profit by determining the probability distribution of the profit per day.
- Assume that the revenue, cost, and number of jobs per day follow specific probability distributions (e.g., normal distribution).
- Use the probability distribution to calculate the probability of making a positive profit (P > 0) on any given day.
The specific probability distribution will depend on the assumptions made about the revenue, cost, and number of jobs per day.
To solve these probability problems, we will use the concept of the binomial distribution.
a) Probability of at least 3 nicks in a 60-meter length:
First, let's find the probability of getting exactly 0, 1, or 2 nicks in a 60-meter length and subtract it from 1 to find the probability of getting at least 3 nicks.
To find the probability of getting exactly k nick(s) in a 60-meter length, we can use the formula for the binomial probability:
P(k) = (n choose k) * p^k * (1-p)^(n-k)
Where n is the total number of trials, p is the probability of success in a single trial, and (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
In this case, n = 60 (the length of wire), k can be 0, 1, or 2 (since we want the complement), and p is the probability of getting a nick in a meter of wire.
Given that there are 20 nicks in a coil of 600-meter wire, the probability of getting a nick in a meter of wire is 20/600 = 1/30.
Therefore, we can calculate the probability of getting exactly k nick(s) in a 60-meter length:
P(0) = (60 choose 0) * (1/30)^0 * (29/30)^(60-0)
P(1) = (60 choose 1) * (1/30)^1 * (29/30)^(60-1)
P(2) = (60 choose 2) * (1/30)^2 * (29/30)^(60-2)
Now, we can calculate the probability of getting at least 3 nicks:
P(at least 3) = 1 - (P(0) + P(1) + P(2))
b) Probability of exactly 4 nicks in a 47-meter length:
Using similar steps as in part (a), we can calculate the probability of getting exactly 4 nicks in a 47-meter length using the formula for the binomial distribution. In this case, n = 47, k = 4, and p = 1/30 (probability of getting a nick in a meter of wire).
P(4) = (47 choose 4) * (1/30)^4 * (29/30)^(47-4)
5. To answer the questions about the driveway sealing business, we need to calculate the probabilities related to the number of customers they get in an evening of soliciting and the profitability of a given day.
a) Probability of getting from 4 to 10 jobs in an evening:
We can use the binomial distribution formula to calculate the probabilities of getting exactly 4, 5, 6, 7, 8, 9, or 10 jobs in an evening. In this case, n (the total number of trials) will vary depending on the time interval they spend ringing doorbells. Let's assume a 3-hour time interval, so n = 3.
P(4 to 10) = P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
b) Probability of making a profit on any given day:
To calculate the probability of making a profit on any given day, we need to consider the revenue (total amount earned) and the costs involved. The revenue is the total amount earned from the number of jobs they get multiplied by the price charged per job. The costs include the daily rental costs for the truck and the spraying equipment, as well as the material cost per driveway.
Let's assume that the number of customers they get follows a Poisson distribution since the average number of customers per hour is given.
We can use the Poisson distribution formula:
P(x) = (e^(-λ) * λ^x) / x!
Where λ is the average number of customers per hour and x is the number of customers.
To calculate the probability of making a profit, we subtract the total cost from the revenue and check if it is greater than zero. If it is, then they made a profit, otherwise, they did not.
Total revenue = (number of customers) * (price charged per driveway)
Total cost = (truck rental cost) + (spraying equipment rental cost) + (material cost per driveway) * (number of customers)
Finally, we divide the number of profitable scenarios by the total number of scenarios to calculate the probability of making a profit.