Question

According to a USA Today Snapshot, drivers say fixing or repaving streets is the best way to make their communities drivable- better than building new roads or adding lanes. \({ }^{14}\) Suppose that \(n=15\) drivers are randomly selected and \(x\) is the number who say that improved road conditions would make their communities more drivable. Let \(p=.4\) when finding probabilities associated with any following outcomes: a. What is the probability distribution for \(x ?\) b. What is \(P(x \leq 4) ?\) c. Find the probability that \(x\) exceeds 5 . d. What is the largest value of \(c\) for which \(P(x \leq c) \leq .5 ?\)

Short answer

Question: In a sample of 15 drivers, there is a 0.4 probability that they believe that improving the road conditions would make their communities more drivable. Find a) the probability distribution for x, b) the probability that x ≤ 4, c) the probability that x exceeds 5, and d) the largest value of c for which P(x ≤ c) ≤ 0.5. Answer: a) The probability distribution for x is calculated using the binomial probability formula: \(P(x) = \binom{15}{x}0.4^x(1-0.4)^{15-x}\) for x values from 0 to 15. b) The probability that x ≤ 4 can be found by adding the probabilities of x = 0, 1, 2, 3, and 4, which can be calculated using the probability distribution formula. c) The probability that x exceeds 5 can be found by summing up the probabilities from x = 6 to x = 15, which can also be calculated using the probability distribution formula. d) The largest value of c for which P(x ≤ c) ≤ 0.5 is the value of x just before the cumulative probability exceeds 0.5, which can be found by calculating the cumulative probabilities for each value of x and identifying the value where the threshold is crossed.

Step-by-step solution

Understanding the binomial probability formula

The binomial probability formula for a given event can be written as: \(P(x) = \binom{n}{x}p^x(1-p)^{n-x}\) Where \(n\) represents the number of trials, \(x\) is the number of successes, \(p\) is the probability of success in a single trial, and \(\binom{n}{x} = \frac{n!}{x!(n-x)!}\) is the number of ways to choose \(x\) successes out of \(n\) trials.

Find the probability distribution for \(x\)

Now, we will compute the probability distribution for \(x\) (the number of drivers). This is essentially a list of probabilities for each possible value of \(x\), from 0 to 15 (\(n\)): \(P(x) = \binom{15}{x}0.4^x(1-0.4)^{15-x}\) Now, list all possible values of \(x\) and their respective probabilities using this formula: x | P(x) ------------- 0 | \(P(0) = \binom{15}{0}0.4^0(1-0.4)^{15}\) 1 | \(P(1) = \binom{15}{1}0.4^1(1-0.4)^{14}\) ... 15 | \(P(15) = \binom{15}{15}0.4^{15}(1-0.4)^{0}\)

Find \(P(x\leq4)\)

We can find the probability that \(x\leq4\) by adding the probabilities of \(x=0,1,2,3,\) and \(4\). Using the probability distribution computed in step 2, we have: \(P(x\leq4) = P(0)+P(1)+P(2)+P(3)+P(4)\) Calculate using the binomial distribution formula, and we obtain the probability.

Find the probability that \(x\) exceeds 5

We can find the probability that \(x\) exceeds 5 by summing up the probabilities from \(x=6\) to \(x=15\): \(P(x>5) = P(6)+P(7)+...+P(15)\) Calculate using the binomial distribution formula, and we obtain the probability.

Find the largest value of \(c\) for which \(P(x\leq c)\leq 0.5\)

To find the largest value of \(c\), we need to find the smallest \(x\) for which the cumulative probability exceeds 0.5. We'll start by calculating the cumulative probabilities for each value of \(x\): \(cumulative\_prob(x) = P(x\leq x) = P(0)+P(1)+...+P(x)\) Starting with \(x=0\), calculate the cumulative probabilities until the value exceeds 0.5. The largest value of \(c\) for which \(P(x\leq c)\leq 0.5\) is the value of \(x\) just before this happens.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the study of random events and outcomes. It's a framework we use to understand and predict how likely certain events are to happen. It distinguishes between deterministic processes, where outcomes are predictable, and probabilistic ones, where outcomes are uncertain.

Understanding probability theory involves some key concepts like sample space, which is the set of all possible outcomes, and events, which are specific outcomes or combinations of outcomes from the sample space. We use probability to quantify the chance of events occurring, assigning values between 0 (impossible event) and 1 (certain event).

• Random Variables: These are variables that take on different values depending on the outcome of a random process. They can be discrete or continuous.
• Probability Distributions: These describe how probabilities are distributed over the values of a random variable.
• Expected Value: This is the mean or average of a random variable's possible values, weighted by their respective probabilities.

Knowing these concepts helps us make sense of data and patterns in various fields, from science to finance, and is foundational to understanding more complex statistical models.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a specific value. In other words, it is the sum of probabilities for all values up to and including a given point. This is particularly useful when we want to understand the likelihood of a variable falling within a certain range.

To calculate cumulative probability, we often use cumulative distribution functions (CDF). A CDF provides the cumulative probability associated with each value a random variable can assume.

• It is useful for assessing risk and making predictions based on probable outcomes.
• Mathematically, if a random variable X has a probability distribution P(X), the cumulative distribution function is given by:

\[ F(x) = P(X \leq x) = \sum_{k=0}^{x} P(k) \]
This formula computes the cumulative probability by adding up individual probabilities. In binomial distributions, as used in this exercise, it becomes essential when determining the likelihood of observing a set number of successes across trials, such as finding \(P(x \leq 4)\). Understanding cumulative probability is crucial for decision-making processes in various contexts, like quality control or financial forecasting.

Binomial Probability Formula

The binomial probability formula is a fundamental concept in probability theory used to find the likelihood of a specific number of successes in a fixed number of trials. It applies when each trial is independent, and there are only two possible outcomes: success or failure.

The formula is:

• \( P(x) = \binom{n}{x} p^x (1-p)^{n-x} \)

Where:

• \(n\) is the total number of trials.
• \(x\) is the number of successful trials.
• \(p\) is the probability of success on a single trial.
• \(\binom{n}{x} = \frac{n!}{x!(n-x)!}\) is the binomial coefficient, showing the number of ways \(x\) successes can occur in \(n\) trials.

The binomial probability formula helps to compute discrete probability distributions, where we are looking at limited, countable outcomes like flipping a coin several times or surveying a group of drivers.
It becomes a powerful tool for solving problems in statistics and real-world scenarios. For example, figuring out the probability of a certain number of people agreeing on improved road conditions being beneficial, just like in the exercise provided. Mastering this formula enables deeper understanding and application in more complex probability problems.