It is claimed that 70% of the cars on the Valley Highway are going faster than 65 miles per hour. A random sample of 12 cars was observed under normal driving conditions with no police car in sight.
(A) What is the probability that all of them were going faster than 65 miles per hour?
(B) What is the probability that fewer than half of them were going over 65 miles per hour?
(A) What is the probability that all of them were going faster than 65 miles per hour?
Well, let's break out our probability calculators and take a look! Now, we know that 70% of the cars on the Valley Highway are going faster than 65mph, so that means the probability of a car going faster than 65mph is 0.7.
Since we have a sample size of 12, we can calculate the probability of all 12 cars going faster than 65mph by multiplying the individual probabilities together. So, the calculation would look like this:
P(all 12 cars going faster than 65mph) = (0.7)^12
Now, get ready for a big reveal...the answer is approximately 0.038. So, there is about a 3.8% chance that all 12 cars are going faster than 65mph.
(B) What is the probability that fewer than half of them were going over 65 miles per hour?
Ah, the odds of fewer than half the cars going over 65mph, now we're talking! Well, to calculate this probability, we need to consider all the possible scenarios where fewer than half of the cars are speed demons.
Now, let's think about it. If we have 12 cars in total, then for fewer than half of them to be going over 65mph, it means we need either 0, 1, 2, 3, 4, 5, or 6 cars to be speed demons.
To calculate the exact probability, we need to calculate the probability of each of these individual scenarios and then sum them up. Now, I could go through all the calculations, but let's just say it's gonna take a lot of math and a lot of time. So, instead, let me show you a little trick...
A simpler way to calculate this probability is to use the complement rule. The complement of fewer than half of the cars going over 65mph is the same as saying more than or equal to half of the cars are speed demons.
So, let's find the probability of more than or equal to half of the cars going over 65mph. This is equivalent to 1 minus the probability of fewer than half of the cars going over 65mph.
P(more than or equal to half of the cars going over 65mph) = 1 - P(fewer than half of the cars going over 65mph)
And there you have it, my friend! By using the complement rule, you can easily find the probability you're looking for. Just remember to grab a calculator and do some number crunching. Good luck!
To calculate the probabilities, we can use the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes in n trials.
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.
- p is the probability of success for a single trial.
- (1-p) is the probability of failure for a single trial.
- n is the number of trials.
Let's solve each part separately:
(A) What is the probability that all 12 cars were going faster than 65 miles per hour?
In this case, we have n = 12, k = 12, and p = 0.70 (probability of a car going faster than 65 mph).
Using the formula, the probability is:
P(X = 12) = C(12, 12) * 0.70^12 * (1-0.70)^(12-12)
Simplifying, we get:
P(X = 12) = 1 * 0.70^12 * 0^0
It's important to note that any number raised to the power of 0 is equal to 1. Therefore:
P(X = 12) = 0.70^12 * 1
Calculating the result:
P(X = 12) = 0.70^12 ≈ 0.0233
So, the probability that all 12 cars were going faster than 65 miles per hour is approximately 0.0233 or 2.33%.
(B) What is the probability that fewer than half of them were going over 65 miles per hour?
In this case, we need to calculate the probabilities for k = 0, k = 1, k = 2, ..., k = 5.
Using the formula for each value of k and summing them up, we get:
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
To calculate each term, we use the same formula as before with n = 12, p = 0.70, and substitute the value of k:
P(X = 0) = C(12, 0) * 0.70^0 * (1-0.70)^(12-0)
P(X = 1) = C(12, 1) * 0.70^1 * (1-0.70)^(12-1)
P(X = 2) = C(12, 2) * 0.70^2 * (1-0.70)^(12-2)
P(X = 3) = C(12, 3) * 0.70^3 * (1-0.70)^(12-3)
P(X = 4) = C(12, 4) * 0.70^4 * (1-0.70)^(12-4)
P(X = 5) = C(12, 5) * 0.70^5 * (1-0.70)^(12-5)
Calculating each term and summing them up gives:
P(X < 6) ≈ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) ≈ 0.0411
So, the probability that fewer than half of the cars are going over 65 miles per hour is approximately 0.0411 or 4.11%.
To find the probabilities, we will use the binomial probability formula. The binomial distribution is appropriate in this case because we have a fixed number of trials (12 cars observed) and each trial (observing a car) is independent, with the same probability of success (car going faster than 65 mph).
Let's calculate the probabilities:
(A) What is the probability that all of them were going faster than 65 miles per hour?
To find the probability that all 12 cars were going faster than 65 mph, we need to calculate the probability of success (P) for each car, which is given as 70% or 0.7.
Using the binomial probability formula:
P(X = k) = nCk * P^k * (1-P)^(n-k)
where:
X = number of successes (12 cars going faster than 65 mph)
k = desired number of successes (12 cars going faster than 65 mph)
n = number of trials (12 cars observed)
P = probability of success (70% or 0.7)
Plugging in the values, we get:
P(X = 12) = 12C12 * (0.7)^12 * (1-0.7)^(12-12)
P(X = 12) = 1 * (0.7)^12 * (0.3)^0
P(X = 12) = (0.7)^12 * 1
P(X = 12) = 0.7^12
Therefore, the probability that all 12 cars were going faster than 65 mph is approximately 0.0282 or 2.82%.
(B) What is the probability that fewer than half of them were going over 65 miles per hour?
We want to find the probability that less than half of the cars (less than 6) were going faster than 65 mph. It means we need to calculate the probabilities for 0, 1, 2, 3, 4, and 5 successes, and then sum them up.
Using the same binomial probability formula, we can calculate each individual probability for each number of successes. Then, we sum them up:
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Plugging in the values and calculating them one by one:
P(X = 0) = 12C0 * (0.7)^0 * (1-0.7)^(12-0)
P(X = 0) = 1 * 1 * 0.3^12
P(X = 0) ≈ 0.000531441
P(X = 1) = 12C1 * (0.7)^1 * (1-0.7)^(12-1)
P(X = 1) = 12 * 0.7 * 0.3^11
P(X = 1) ≈ 0.006837
P(X = 2) = 12C2 * (0.7)^2 * (1-0.7)^(12-2)
P(X = 2) = 66 * 0.7^2 * 0.3^10
P(X = 2) ≈ 0.04515457
P(X = 3) = 12C3 * (0.7)^3 * (1-0.7)^(12-3)
P(X = 3) = 220 * 0.7^3 * 0.3^9
P(X = 3) ≈ 0.16926303
P(X = 4) = 12C4 * (0.7)^4 * (1-0.7)^(12-4)
P(X = 4) = 495 * 0.7^4 * 0.3^8
P(X = 4) ≈ 0.36379788
P(X = 5) = 12C5 * (0.7)^5 * (1-0.7)^(12-5)
P(X = 5) = 792 * 0.7^5 * 0.3^7
P(X = 5) ≈ 0.36379788
Adding all the individual probabilities:
P(X < 6) ≈ 0.000531441 + 0.006837 + 0.04515457 + 0.16926303 + 0.36379788 + 0.36379788
P(X < 6) ≈ 0.94938179
Therefore, the probability that fewer than half of the observed cars were going over 65 mph is approximately 0.9494 or 94.94%.