The probability is 0.35 that a traffic fatality involves an intoxicated or alcohol-impaired driver or nonoccupant. In ten traffic fatalities, find the probability that the number, Y, which involve an intoxicated or alcohol-impaired driver or nonoccupant is a. exactly three; at least three; at most three.
b. between two and four, inclusive.
c. Find and interpret the mean of the random variable Y. d. Obtain the standard deviation of Y.
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To find the probabilities for the number of traffic fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant, we will use the binomial distribution.
a. For exactly three fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant:
The probability of success (p) is 0.35, and the number of trials (n) is 10. Using the binomial probability formula, we can calculate P(Y = 3):
P(Y = 3) = (10 choose 3) * (0.35^3) * (1 - 0.35)^(10 - 3)
b. For at least three fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant:
We need to calculate P(Y ≥ 3), which is the sum of probabilities for Y = 3, Y = 4, ..., Y = 10. Mathematically, it can be written as:
P(Y ≥ 3) = P(Y = 3) + P(Y = 4) + ... + P(Y = 10)
c. For at most three fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant:
We need to calculate P(Y ≤ 3), which is the sum of probabilities for Y = 0, Y = 1, Y = 2, and Y = 3. Mathematically, it can be written as:
P(Y ≤ 3) = P(Y = 0) + P(Y = 1) + P(Y = 2) + P(Y = 3)
To calculate the mean of the random variable Y (expected value), we multiply the number of trials (n) by the probability of success (p):
Mean (µ) = n * p
To calculate the standard deviation of the random variable Y, we use the formula:
Standard Deviation (σ) = √(n * p * (1 - p))
Now, let's calculate the probabilities and the mean and standard deviation:
a. Probability of exactly three fatalities:
P(Y = 3) = (10 choose 3) * (0.35^3) * (1 - 0.35)^(10 - 3)
b. Probability of between two and four (inclusive) fatalities:
P(Y = 2) + P(Y = 3) + P(Y = 4)
c. Probability of at most three fatalities:
P(Y = 0) + P(Y = 1) + P(Y = 2) + P(Y = 3)
d. Mean (expected value):
Mean (µ) = n * p
e. Standard Deviation:
Standard Deviation (σ) = √(n * p * (1 - p))
To solve this problem, we can use the binomial probability formula. The formula for the probability of Y successes in n trials, where the probability of success for each trial is p, is:
P(Y = k) = (n choose k) * p^k * (1-p)^(n-k)
Let's solve each part of the problem step by step:
a. To find the probability that exactly three traffic fatalities involve an intoxicated or alcohol-impaired driver or nonoccupant, we can use the formula above with n = 10, k = 3, and p = 0.35:
P(Y = 3) = (10 choose 3) * 0.35^3 * (1-0.35)^(10-3)
To calculate "10 choose 3," which denotes the number of ways to choose 3 items out of 10, we can use the combination formula:
(10 choose 3) = 10! / (3! * (10-3)!)
where "!" denotes the factorial function.
b. To find the probability that the number of traffic fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant is between two and four, inclusive, we calculate the probabilities for Y = 2, 3, and 4, and then sum them:
P(Y = 2) + P(Y = 3) + P(Y = 4) = [(10 choose 2) * 0.35^2 * (1-0.35)^(10-2)] + [(10 choose 3) * 0.35^3 * (1-0.35)^(10-3)] + [(10 choose 4) * 0.35^4 * (1-0.35)^(10-4)]
c. The mean of a binomial random variable can be calculated using the formula:
μ = n * p
where μ represents the mean, n is the number of trials, and p is the probability of success for each trial. In this scenario, μ represents the average number of traffic fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant out of 10 fatalities.
μ = 10 * 0.35
d. The standard deviation of a binomial random variable can be calculated using the formula:
σ = sqrt(n * p * (1-p))
where σ represents the standard deviation, n is the number of trials, and p is the probability of success for each trial. In this scenario, σ represents the standard deviation for the number of traffic fatalities involving an intoxicated or alcohol-impaired driver or nonoccupant out of 10 fatalities.
σ = sqrt(10 * 0.35 * (1-0.35))
By calculating each part of the problem using the formulas and values provided, you can find the requested probabilities and statistical measures.