Question

Use Table 1 in Appendix I to find the following: a. \(P(x<12)\) for \(n=20, p=.5\) b. \(P(x \leq 6)\) for \(n=15, p=.4\) c. \(P(x>4)\) for \(n=10, p=.4\) d. \(P(x \geq 6)\) for \(n=15, p=.6\) e. \(P(3

Short answer

Answer: To find the probability of having more than 3 but less than 7 successes in 10 trials with a success probability of 0.5, we can use a binomial probability table (like Appendix I mentioned in the solution). Referring to the table and following the steps in the solution, we have to locate the row for n=10 trials and p=0.5 and sum the probabilities for x=4-6. This is the total probability of having more than 3 but less than 7 successes in 10 trials.

Step-by-step solution

Find the relevant probabilities in the table for n=20 and p=.5

In Appendix I, locate the row for n=20 trials with probability p=.5. We find that the probability of 0-11 successes is given in the table.

Add up the probabilities as given in the table

Sum the probabilities for x=0-11 to find P(x<12). The total probability is the probability that we have less than 12 successes in 20 trials. #b: P(x≤6) for n=15, p=.4#

Find the relevant probabilities in the table for n=15 and p=.4

In Appendix I, locate the row for n=15 trials with probability p=.4. We find that the probability of 0-6 successes is given in the table.

Add up the probabilities as given in the table

Sum the probabilities for x=0-6 to find P(x≤6). The total probability is the probability that we have 6 or fewer successes in 15 trials. #c: P(x>4) for n=10, p=.4#

Find the relevant probabilities in the table for n=10 and p=.4

In Appendix I, locate the row for n=10 trials with probability p=.4. We find that the probabilities of 5-10 successes are given in the table.

Add up the probabilities as given in the table

Sum the probabilities for x=5-10 to find P(x > 4). The total probability is the probability that we have more than 4 successes in 10 trials. #d: P(x ≥ 6) for n=15, p=.6#

Find the relevant probabilities in the table for n=15 and p=.6

In Appendix I, locate the row for n=15 trials with probability p=.6. We find that the probabilities of 6-15 successes are given in the table.

Add up the probabilities as given in the table

Sum the probabilities for x=6-15 to find P(x≥6). The total probability is the probability that we have 6 or more successes in 15 trials. #e: P(3<x<7) for n=10, p=.5#

Find the relevant probabilities in the table for n=10 and p=.5

In Appendix I, locate the row for n=10 trials with probability p=.5. We find that the probabilities of 4-6 successes are given in the table.

Add up the probabilities as given in the table

Sum the probabilities for x=4-6 to find P( 3< x<7). The total probability is the probability that we have more than 3 but less than 7 successes in 10 trials.

Key concepts

Probability Theory

Probability theory is the mathematical framework that allows us to analyze random events and quantify the likelihood of various outcomes. At the heart of this theory is the concept of a probability distribution, which gives a complete description of the probability of each possible outcome in a random experiment.

For instance, when flipping a fair coin, there are two possible outcomes—heads or tails—with each having a probability of 0.5. But in more complex situations, like the binomial probability distribution used in the provided exercise, we deal with experiments that have two possible outcomes (often termed 'success' and 'failure') over several trials.

In the binomial setting, the probability of observing a certain number of successes in 'n' trials is determined by the parameters 'n' (the number of trials) and 'p' (the probability of success on any given trial). The exercises given are practical applications of probability theory, where the objective is to use statistical tables to calculate the likelihood of certain outcomes given these parameters.

Statistical Methodologies

Statistical methodologies involve techniques that are used to collect, summarize, analyze, and interpret numerical data and make decisions based on that data. One such technique is using probability tables, like Table 1 in Appendix I referenced in the exercise, which is integral in solving problems associated with binomial distributions without resorting to complex mathematical computations each time.

These tables consolidate the results of the binomial formula for various probabilities and number of trials (n), which greatly simplifies the process of finding cumulative probabilities—like 'the probability of x or fewer successes', or 'the probability of more than x successes'.

Using this methodology, we avoid the laborious task of manually calculating binomial probabilities and instead make efficient use of already computed values, demonstrating how statistical tools enhance and streamline the problem-solving process in probability and statistics.

Probability Distribution Analysis

Probability distribution analysis is a critical aspect of understanding the behavior of random variables. It involves studying the properties and characteristics of probability distributions. In the context of the binomial probability distribution, analysis involves determining the likelihood of a given number of successes within a predefined number of trials, considering a consistent probability of success.

The steps outlined in the exercise demonstrate how to perform such an analysis using a probability distribution table. One common measure we can determine is the cumulative probability, which is the probability that a random variable is less than or equal to a certain value. For example, we calculate this by adding the probabilities of all outcomes less than or equal to the specified value. Furthermore, the binomial distribution being discrete allows us to calculate the probability for an interval, like finding the probability that the random variable falls strictly between two values, as in exercise part e. P(3
Understanding and analyzing binomial distributions is fundamental for students, as it is routinely applied across fields such as finance, health sciences, and any domain that requires decision-making under uncertainty.