Using Probabilities for Significant Eventsa. Find the probability...

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

1:27 minutes

Problem 5.1.20a

Textbook Question

Using Probabilities for Significant Events

a. Find the probability of getting exactly 1 match.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability of getting exactly 1 match in a given scenario. This typically involves a probability distribution, such as the binomial distribution, where we calculate the likelihood of a specific number of successes (matches) in a series of trials.

Step 2: Identify the parameters of the problem. Determine the total number of trials (n), the probability of success in a single trial (p), and the number of successes (k) we are interested in. For this problem, k = 1 since we are looking for exactly 1 match.

Step 3: Use the binomial probability formula to calculate the probability of exactly 1 match. The formula is:

P(X = k) = C(n, k) * p

^k * (1 - p)^n-k, where C(n, k) is the number of combinations and can be calculated as

C(n, k) = n! / (k! * (n - k)!)

Step 4: Substitute the values of n, k, and p into the formula. Compute the combination term

C(n, k)

, the probability term

p

^k, and the complement term

(1 - p)

^n-k. Multiply these together to find the probability.

Step 5: Interpret the result. The calculated probability represents the likelihood of getting exactly 1 match in the given scenario. Ensure the result makes sense in the context of the problem and check for any errors in the calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. A probability of 0 indicates that the event cannot happen, while a probability of 1 indicates certainty. In the context of finding the probability of getting exactly one match, it involves calculating the chances based on the total number of trials and the desired outcome.

Binomial Distribution

The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). This concept is essential for calculating the probability of getting exactly one match in scenarios where there are multiple attempts.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It is crucial for determining the number of ways to choose a specific number of successes from a larger set. In the context of finding the probability of exactly one match, combinatorial calculations help in identifying how many different ways one match can occur among the total trials.

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Using Probabilities for Significant Eventsa. Find the probability of getting exactly 1 match.