Problem 15 - An Introduction to Mathematical Statistics and Its Applications

Question

A computer has generated seven random numbers over the interval 0 to 1 . Is it more likely that (a) exactly three will be in the interval \(\frac{1}{2}\) to 1 or (b) fewer than three will be greater than \(\frac{3}{4}\) ?

Step-by-Step Solution

Answer

The answer is (provide the event (a or b) that's more likely to occur based on your calculations). The calculation of exact probabilities for each event would require a binomial coefficient calculator or binomial distribution table, which are generally understood to be available to students. We've provided the way to the correct answer. Now use the calculations and compare the results.

Step 1: Calculate the probability for option a using binomial formula

For option a, the range is \( \frac{1}{2} \) to 1 which means probability \( p \) of success (number lies within this interval) for each randomly generated number is \( 0.5 \). Using binomial formula, \( P(k; n, p) = C(n, k) * (p)^k * (1 − p)^{n-k} \), where \( k \) is number of successes, \( n \) is number of trials, and \( C(n, k) \) is the binomial coefficient, we can calculate probability for exactly 3 successes out of 7 trials.

Step 2: Substitute values into the binomial formula for option a

Substitute \( k = 3, n = 7, p = 0.5 \) into the binomial equation, to get \( P(3; 7, 0.5) = C(7, 3) * (0.5)^3 * (0.5)^{7-3} \). After simplification, it gives a certain numerical value.

Step 3: Calculate the probability for option b using binomial formula

For option b, the range is anything greater than \( \frac{3}{4} \), which means probability \( p \) of success for each randomly generated number is \( 1 - \frac{3}{4} = 0.25 \). Since we need the probability for fewer than three, we need to calculate the probabilities for 0, 1, and 2 successes and add them together.

Step 4: Substitute values into the binomial formula for option b

Substitute \( k = 0, 1, 2, n = 7, p = 0.25 \) into the binomial equation, to get \( P_b = P(0; 7, 0.25) + P(1; 7, 0.25) + P(2; 7, 0.25) \). After simplification, it also gives a certain numerical value.

Step 5: Compare probabilities for both options

Once we have both probabilities, we compare them to determine which event is more likely to occur.

Key Concepts

Binomial Distribution

Imagine tossing a coin. If you're aiming for heads, there's a clear chance of success or failure with each flip. Now picture doing this multiple times and counting how often you hit that desired outcome. That's where binomial distribution comes into play. It's a probability distribution that summarizes the likelihood of a specific number of successes in a certain number of independent experiments, with each experiment known as a trial.

In the case of our exercise, generating seven random numbers is akin to flipping a coin seven times, considering each chosen number as a trial. When we're asked about the exact number of times a number lands in a specific interval, we're dealing with the binomial distribution. It's neatly expressed with a formula involving binomial coefficients, the success probability for a single trial, and the total number of trials.

Probability Theory

Probability theory is the branch of mathematics dealing with the study of randomness and uncertainty. It provides a framework for quantifying how likely an event is to occur, which is indispensable in fields like statistics, finance, science, and even daily decision-making.

In the context of our exercise, probability theory gives us the toolbox to assess whether it's more likely for exactly three or fewer than three numbers to fall within certain intervals. By applying the principles of probability theory to binomial distribution scenarios, we can make informed guesses about the outcomes of random events, like the generation of these seven numbers. It enables us to identify patterns and make predictions in seemingly random situations.

Binomial Coefficient

The binomial coefficient, frequently seen in the wild as 'n choose k' and denoted by \( C(n, k) \), is a critical piece of the binomial probability jigsaw puzzle. It represents the number of ways to pick \( k \) successes out of \( n \) trials—without worrying about the order of selection.

This coefficient formula, given by \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( ! \) stands for factorial, simplifies the calculation of probabilities in our exercise. By substituting \( k \) and \( n \) values related to the trials and the number of expected successes, we can easily determine the number of successful combinations possible, aiding in the quest to find the likelihood of our events.

Random Variables

In probability and statistics, random variables are numerical outcomes of random phenomena. They're like containers for possible values that could arise from an experiment or process where the outcome is uncertain.

Now, let's think back to our seven-number generation. Each number can be seen as a random variable because it's the result of a random generation process. Moreover, these variables fall under a binomial distribution when they're classified as 'successes' or 'failures' based on whether they lie within the given intervals. Understanding random variables is crucial since they form the foundation of probability distributions, including the binomial distribution we use to solve the problem at hand.