The National Safety Council publishes information about...

The National Safety Council publishes information about automobile accidents in Accident Facts. According to that document, the probability is 0.40 that a traffic fatality will involve an intoxicated or alcoholimpaired driver or nonoccupant. In eight traffic fatalities, find the probability that the number, $Y,$ that 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 $Y.$

The National Safety Council publishes information about automobile accidents in Accident Facts. According to that document, the probability is 0.40 that a traffic fatality will involve an intoxicated or alcoholimpaired driver or nonoccupant. In eight traffic fatalities, find the probability that the number, $Y,$ that 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 $Y.$

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success on a single trial). This distribution is used when there are only two possible outcomes in each trial, typically considered as 'success' or 'failure'.

Probability Mass Function (PMF)

The Probability Mass Function for a binomial distribution gives the probability of observing exactly k successes in n independent trials. It is defined by the formula P(Y = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) is a binomial coefficient that counts the number of ways to choose k successes from n trials. This function is fundamental for calculating exact probabilities for specific outcomes.

Cumulative Probability

Cumulative probability involves summing the probabilities of several outcomes to find the likelihood that a random variable falls within a specified range. For a binomial distribution, this could mean summing the PMF over a range of k values to compute the probability of at most, at least, or a range of successes. This approach is useful for answering questions that ask for probabilities of events such as ‘at most three’ or ‘at least three’ successes.

Mean of a Binomial Distribution

The mean (or expected value) of a binomial distribution is calculated using the formula ? = n * p. This value represents the average number of successes you would expect if you repeated the experiment many times. It provides a measure of the central tendency of the distribution, indicating the typical outcome in the long run.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution is a measure of the spread or variability around the mean. It is computed using the formula ? = ?(n * p * (1-p)). This statistic quantifies the average amount by which the number of successes deviates from the mean, giving insight into the distribution's dispersion.

Transcript

00:02 In this exercise, we are given that the probability is 0 .4, that a traffic fatality will involve an intoxicated or alcohol impaired driver or non -occupant.

00:13 In the first part of the problem a, we're supposed to find the probability that the number y of fatalities that involve intoxicated or alcohol impaired drivers or non -occupant is exactly 3.

00:30 At least three and at most three.

00:34 Now, before we can do that, we would need to create the probability distribution for that random variable y.

00:43 And first, y can take on the value zero all the way to 8.

00:51 Because we are looking at 8 traffic fatalities.

00:58 Write the numbers all the way to 8.

01:01 And then the next column will be for the probabilities.

01:11 You need to fill up the table with the different probabilities for different numbers.

01:15 So we use a binomial distribution formula.

01:21 The number of fatalities is 0 out of 8 trials and the success probability is 0 .4.

01:39 And we say false.

01:43 From there we copy the formula all the way through for the 8.

01:47 Different values of y.

01:50 Now we have our probabilities, probability distribution and in part a of the question we are looking at three different probabilities the first one the probability that y is exactly three that means that we're looking at y equals three and come back to the table the value is zero points so you just need to put that there 0 .278 which you can round off to make it 0 .279.

02:45 Next we're looking at the probability that the number is at least three.

02:53 So at least three means it could be three, four or four or five, six, six, 7 and 8.

03:04 So the probability that y is at least 3 will be obtained as follows.

03:26 You will have to get the probabilities for 0, 1 and 2.

03:36 Then we add them up.

03:39 And then we subtract from 1 the sum of these probabilities.

03:45 So it will be equal to 1 minus the sum...

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