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A probability distribution is a powerful concept in statistics that shows all the possible outcomes of a random experiment and the probability of each outcome. In the context of tailgating on the highway, we use a **binomial distribution** because there are two possible outcomes for each of the 12 vehicles: either they tailgate or they do not.

The binomial distribution helps us determine the probability of a certain number of vehicles tailgating. The binomial probability formula is given as: \[ P(r) = \binom{n}{r} p^r (1-p)^{n-r} \] where:

• \( n \) is the total number of vehicles (in our case, 12).
• \( r \) is the number of vehicles tailgating.
• \( p \) is the probability of tailgating for each vehicle (0.35).
• \( 1-p \) is the probability of not tailgating (0.65).

By calculating the probability for each \( r \) (0 through 12), we form a complete probability distribution, showcasing the likelihood of each possible number of vehicles tailgating.

The concept of expected value can be thought of as the average outcome we expect in a probabilistic scenario. It's like predicting the most "usual" case outcome.

In our highway scenario, the expected number of vehicles that tailgate is calculated by multiplying the total number of vehicles \( n \) by the probability of a single vehicle tailgating \( p \). This is done through the formula: \[ E(r) = n \cdot p \] For our case: \[ E(r) = 12 \cdot 0.35 = 4.2 \] This tells us that out of 12 cars passing by, we can "expect" about 4.2 vehicles to tailgate, on average. This doesn't mean you can have 0.2 of a vehicle, but rather that in many repeated scenarios, the average approaches this number.

Standard deviation is a key measure that tells us how much variation or "spread" exists from the average or expected value in our data. In simpler terms, it shows how spread out the tailgating behavior is among the 12 vehicles.

For a binomial distribution, the standard deviation \( \sigma \) is found using the formula: \[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \] Substituting in our values:\[ \sigma = \sqrt{12 \cdot 0.35 \cdot 0.65} \] By calculating this, we find the standard deviation, which quantifies the variation in the number of vehicles that tailgate. A smaller standard deviation means the results are close to the expected value, while a larger one suggests more variety in outcomes.

A histogram is a graphical representation of data that uses bars to show the frequency or probability of different possible outcomes of a dataset. In statistics, it is particularly useful to understand the probability distribution of a random variable, such as tailgating in this example.

To create the histogram:

• The x-axis will represent the random variable \( r \), the number of vehicles tailgating, ranging from 0 to 12.
• The y-axis will show the probability of each of these outcomes, calculated using the binomial formula.

Each bar on the histogram represents the probability of that particular number of vehicles tailgating, giving us a clear visual of how likely each scenario is. Seeing this distribution in a histogram format makes it easier to understand which outcomes are more probable and how the data is spread.