Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

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1:51 minutes

Problem 5.1.20b

Textbook Question

Using Probabilities for Significant Events

b. Find the probability of getting 1 or fewer matches.

Verified step by step guidance

Step 1: Identify the type of probability distribution involved in the problem. If the problem involves matches (e.g., successes in trials), it is likely a binomial distribution. Confirm the number of trials (n) and the probability of success (p) for each trial.

Step 2: Define the event of interest. In this case, the event is 'getting 1 or fewer matches,' which means calculating the cumulative probability for 0 matches and 1 match.

Step 3: Use the probability mass function (PMF) of the binomial distribution to calculate the probabilities for 0 matches and 1 match. The PMF is given by:

P (X = k) = \frac{n!}{k! (n - k)!} \cdot p^{k} \cdot (^{1 - p) n - k}

, where k is the number of matches.

Step 4: Add the probabilities for 0 matches and 1 match to find the cumulative probability. This is expressed as:

P (X \leq 1) = P (X = 0) + P (X = 1)

Step 5: Use a calculator or statistical software to compute the individual probabilities and their sum. Alternatively, if a cumulative distribution function (CDF) is available, use it directly to find

P (X \leq 1)

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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 significant events, understanding how to calculate probabilities helps in assessing the chances of various outcomes.

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 particularly useful for scenarios where there are two possible outcomes, such as success or failure. In this case, finding the probability of getting 1 or fewer matches can be calculated using the binomial formula.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. It is calculated by summing the probabilities of all outcomes up to that value. For the question at hand, determining the cumulative probability of getting 1 or fewer matches involves adding the probabilities of getting 0 matches and 1 match.

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Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).