Probability Concepts and Experiments

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Questions and Answers

Which two events are mutually exclusive? Explain why.

  • B and C
  • A and D
  • A and C (correct)
  • A and B

Events A and C are complementary events, meaning they are exact opposites of each other.

False (B)

Which two events are independent of each other? Explain why.

  • A and C
  • B and D
  • A and B
  • A and D (correct)

What is the formula to calculate the probability of an event using the frequentist approach?

<p>The probability of an event is equal to the number of times the event occurred divided by the total number of times the event was attempted.</p> Signup and view all the answers

Explain why buying 80 scratch-off lottery tickets can be considered a binomial experiment.

<p>Each ticket purchase represents an independent trial with two possible outcomes: winning or losing. The probability of winning remains constant for each ticket, and the number of tickets purchased is fixed. These characteristics define a binomial experiment.</p> Signup and view all the answers

What is the mean of the resulting binomial distribution for the scratch-off lottery tickets example?

<p>The mean of the binomial distribution is 16, meaning on average, 16 out of the 80 tickets will be expected to win.</p> Signup and view all the answers

Why is it reasonable to assume that the binomial distribution in the scratch-off lottery tickets example is approximately normal?

<p>The conditions for approximating a binomial distribution with a normal distribution are met: np = 16 and n(1-p) = 64, both of which are greater than or equal to 10.</p> Signup and view all the answers

What is the standard deviation of the binomial distribution in the scratch-off lottery tickets example?

<p>The standard deviation of the binomial distribution is roughly 3.6.</p> Signup and view all the answers

Is it unusual for 20 of the 80 tickets to win? Explain your reasoning using a z-score.

<p>No, because the z-score is 1.115, which is less than 2. (B)</p> Signup and view all the answers

What is the probability that a subject's startle response will be less than 50.3 milliseconds?

<p>The probability that a subject's startle response will be less than 50.3 milliseconds is about 84%, as it represents 84% of the area under the normal curve, including the 50% below the mean and an additional 34% between the mean and 50.3.</p> Signup and view all the answers

What is the corresponding startle response for a person in this study with a z-score of 1.28?

<p>The corresponding startle response for a person with a z-score of 1.28 is 53.8 milliseconds.</p> Signup and view all the answers

Flashcards

Frequentist probability

Estimating probability by conducting experiments and observing the results. It's based on repeated trials.

Mutually Exclusive Events

Events that cannot occur at the same time.

Complementary Events

Events that are exact opposites and their probabilities add to 1.

Independent Events

Events where the outcome of one does not affect the outcome of another.

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Dependent Events

Events where the outcome of one affects the outcome of another.

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Binomial Experiment

A series of identical trials with two possible outcomes (success/failure) each time.

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Binomial Distribution Mean

The expected average number of successes in a binomial experiment.

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Binomial Distribution Standard Deviation

A measure of spread in a binomial distribution.

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Approximately Normal Distribution

A distribution that resembles a normal bell curve, when certain conditions are met (np and nq are at least 15).

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Z-score

A way to measure how many standard deviations a value is from the mean.

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Normal Distribution

A bell-shaped probability distribution where data clusters around the mean.

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Empirical Rule

A rule of thumb stating that for a normal distribution, 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations of the mean.

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Percentile

The value below which a given percentage of observations in a dataset falls.

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Standard Deviation

A measure of the dispersion of a dataset relative to its mean.

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90th percentile

The value that separates the top 10% of data from the bottom 90% in a dataset.

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Mean

The average of a dataset.

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Probability

The likelihood of an event occurring.

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Study Notes

Activity #4 - Solutions

  • Group Member Names and Music Preferences: Record names and favorite musician/band for each group member.

Frequentist Probability

  • Estimating Probability (Thumbtack): Conduct an experiment to determine the probability of a thumbtack landing point-up. Count the number of times the thumbtack lands point up and divide by the total number of drops to estimate the probability.

Probability Basics

  • Mutually Exclusive Events: Events that cannot occur simultaneously. In this specific case, event A (going to class) and event C (skipping class) are mutually exclusive.
  • Complementary Events: Two events are complementary if they cover all possible outcomes. However events A (going to class) and C(skipping class) are not complementary.
  • Independent Events: The probability of one event occurring is unaffected by the occurrence of another event. Examples of independent events from this activity are D (Biden being president) and A (going to class).
  • Dependent Events: The probability of one event is affected by the occurrence of another event. An example of dependent events from this activity is B (rain) and A(going to class). If it rains, it may be less likely that you will go to class.

Binomial Experiment

  • Explanation: A binomial experiment involves a fixed number of identical trials, where each trial has only two possible outcomes (success or failure). Example from this activity: scratch-off tickets and whether each wins or loses something.
  • Parameters: n (number of trials), p (probability of success on a single trial).

Binomial Distribution

  • Mean: The mean (expected value) of a binomial distribution is calculated as n * p. In this case, the mean (expected number of winning scratch-off tickets) is 80 * 0.2 = 16.
  • Standard Deviation: The standard deviation of a binomial distribution is calculated as the square root of n * p * (1 - p). In this case, the standard deviation is approximately 3.6.
  • Approximation with Normal Distribution: A binomial distribution can often be approximated by a normal distribution when n * p and n * (1 - p) are both greater than or equal to 5.
  • Empirical Rule: The Empirical Rule can be used to estimate the probabilities in the normal distribution, particularly within one, two, or three standard deviations of the mean.

Startle Response and Normal Distribution

  • Probability and Normal Distribution: Understanding probabilities for events, including the probability of an individual's startle response being between a certain range.
  • Means and Standard Deviations: Calculating the mean and standard deviation of a normally distributed variable as well as interpretation.

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Math 1240 Activity #4 Fall 2024 PDF

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