Discrete Probability Distribution Quiz
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Questions and Answers

What defines a discrete random variable?

  • Its value is determined by a random experiment. (correct)
  • It can take an infinite number of values.
  • It has a continuous probability distribution.
  • It is always a whole number.

    • In the context of the coin flipping example, what is represented by the random variable X?

  • The number of coins flipped.
  • The number of tails showing.
  • The number of heads showing. (correct)
  • The combined weight of the coins.

    • Which of the following is NOT a requirement for a probability distribution?

  • All probabilities must be between 0 and 1.
  • Each probability must be non-negative.
  • The sum of the probabilities must equal 1.
  • The average of the probabilities must equal the mode. (correct)

    • What is the probability of rolling a sum of 7 with two dice?

    <p>1/6</p> Signup and view all the answers

    How many outcomes are possible when flipping three coins?

    <p>8</p> Signup and view all the answers

    In the discrete probability distribution for rolling two dice, what is the probability of rolling a sum of 2?

    <p>1/36</p> Signup and view all the answers

    Which outcome would NOT be part of the sample space for the number of heads when flipping three coins?

    <p>4 heads</p> Signup and view all the answers

    What is the probability of getting exactly 3 heads when flipping three coins?

    <p>1/8</p> Signup and view all the answers

    Which of the following options is NOT considered a probability distribution?

    <p>Distribution with negative probabilities</p> Signup and view all the answers

    What is the formula for calculating the population mean?

    <p>$ rac{ΣX}{N}$</p> Signup and view all the answers

    What is the mean of a discrete random variable defined as $μ = ΣX ∙ P(X)$?

    <p>The average outcome of the variable</p> Signup and view all the answers

    In the example of tossing three coins, what is the calculated mean number of heads?

    <p>1.5</p> Signup and view all the answers

    According to the rounding rule for the mean, variance, and standard deviation, how many decimal places should results be rounded to?

    <p>One more decimal place than the outcome X</p> Signup and view all the answers

    When calculating the mean from the probability distribution, which calculation is correct?

    <p>$Σ(X ∙ P(X))$</p> Signup and view all the answers

    If a probability distribution has a negative probability value, what can be concluded?

    <p>It is not a probability distribution</p> Signup and view all the answers

    What does the notation $ΣX ∙ P(X)$ signify in the context of probability?

    <p>The sum of the products of outcomes and probabilities</p> Signup and view all the answers

    What is the mean number of trips based on the provided probability distribution?

    <p>1.23</p> Signup and view all the answers

    How is the variance of a probability distribution calculated?

    <p>By summing the square of each outcome multiplied by its probability and subtracting the square of the mean</p> Signup and view all the answers

    What does the symbol E(X) represent in the context of a probability distribution?

    <p>The expected value of a discrete random variable</p> Signup and view all the answers

    Given the variance computation, what is the relationship of the standard deviation to the variance?

    <p>Standard deviation is the square root of the variance</p> Signup and view all the answers

    Using the outcomes 1 through 6 with uniform probability, what is the variance?

    <p>2.91</p> Signup and view all the answers

    What do the probabilities in the distribution represent?

    <p>The likelihood of each specific number of trips occurring</p> Signup and view all the answers

    If a probability distribution has outcomes and their probabilities, which outcome occurs most frequently based on the example given?

    <p>1 trip</p> Signup and view all the answers

    How can expectation be applied in real-world scenarios?

    <p>To assess theoretical averages in decision-making</p> Signup and view all the answers

    What is the total number of possible outcomes represented in the given scenario?

    <p>32</p> Signup and view all the answers

    What is the probability of having exactly 2 girls in the given outcomes?

    <p>10/32</p> Signup and view all the answers

    In the distribution of girls (0 to 5), how many girls correspond to the probability of 1/32?

    <p>0 girls</p> Signup and view all the answers

    How many outcomes correspond to having 1 girl in the scenario?

    <p>5</p> Signup and view all the answers

    What probability corresponds to having 4 girls in the outcomes?

    <p>5/32</p> Signup and view all the answers

    If the outcomes are distributed among 5 children, what is the range for the number of girls?

    <p>0 to 5</p> Signup and view all the answers

    Which probability corresponds to having exactly 3 boys?

    <p>10/32</p> Signup and view all the answers

    How many outcomes have an equal number of boys and girls?

    <p>10</p> Signup and view all the answers

    What is the mean number of spots that appear when a die is tossed?

    <p>3.5</p> Signup and view all the answers

    If the outcomes of a die are represented by 1, 2, 3, 4, 5, and 6, what is the probability of rolling a 4?

    <p>1/6</p> Signup and view all the answers

    In a family of five children, if the mean number of girls is represented by X, what is the expected value of X if the probability of having a girl is 0.5?

    <p>3</p> Signup and view all the answers

    Using the formula µ = 𝜮𝑿 ∙ 𝑷(𝑿), what does µ represent?

    <p>The mean of the random variable</p> Signup and view all the answers

    What is the total of the outcomes when calculating the mean for a fair die?

    <p>21</p> Signup and view all the answers

    In the calculation of the mean for five children's genders (M for male, F for female), what outcome represents the scenario where there are no girls?

    <p>MMMMM</p> Signup and view all the answers

    What would be the result of rolling a die and getting a sum of 15 on three successive rolls?

    <p>This is not possible.</p> Signup and view all the answers

    If the expected number of girls in a family with five children is 2.5, which choice correctly represents potential outcomes?

    <p>0, 1, 2, 3, 4, or 5 girls</p> Signup and view all the answers

    What is the expected value of the gain when purchasing one ticket for $1 in the color television lottery?

    <p>$-0.65</p> Signup and view all the answers

    In a binomial experiment, which of the following accurately describes the concept of 'n'?

    <p>The total number of independent trials</p> Signup and view all the answers

    What is the proper formula for calculating the variance of a binomial random variable?

    <p>$n imes p imes q$</p> Signup and view all the answers

    In the example provided, what was the probability of Faith getting at least one hit in three official times at bat?

    <p>0.692</p> Signup and view all the answers

    What does the variable 'p' represent in a binomial random variable context?

    <p>The probability of success in a trial</p> Signup and view all the answers

    Which of the following statements about a binomial experiment is correct?

    <p>There are only two possible outcomes for each trial.</p> Signup and view all the answers

    What is calculated by the formula $P(X) = nCr imes p^r imes q^{n-r}$?

    <p>Probability distribution of a binomial random variable</p> Signup and view all the answers

    If the probability of success p in a trial is 0.4, what is the probability of failure q?

    <p>0.6</p> Signup and view all the answers

    Study Notes

    Discrete Probability Distribution

    • A random variable is a variable whose value is determined by a random experiment.
    • A discrete probability distribution is a table of formulas that lists the probabilities for each outcome of a random variable, X.
    • The sum of the probabilities of all the events in the sample space must equal 1 (ΣP(X) = 1).
    • The probability of each event in the sample space must be between or equal to 0 and 1 (0 ≤ P(X) ≤ 1).

    Random Variable

    • A variable whose value is determined by a random experiment.

    Example 1: Flipping Three Coins

    • Possible Outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
    • Number of Heads (x): 3, 2, 2, 1, 2, 1, 1, 0
    • Probability Distribution: P(X=0) = 1/8, P(X=1) = 3/8, P(X=2) = 3/8, P(X=3) = 1/8

    Example 2: Sum of Rolling Two Dice

    • Outcomes of Rolling Two Dice: Each combination of outcomes for rolling two dice is listed. There are 36 possible outcomes in total.
    • Probability Distribution: Shows probabilities for each sum from rolling two dice e.g. P(X=2) = 1/36, P(X=3) = 2/36, ... P(X=12) = 1/36

    Example 3: Determining if a Distribution is Probability Distribution

    • The sum of the probabilities must add up to 1.
    • The values for P(X) should be between 0 and 1.
    • Examples of distributions that are not probability distributions included.

    Quantitative Measures of Random Variable: Mean

    • Sample Mean: X = ΣX / n
    • Population Mean: μ = ΣX / N

    Quantitative Measures of Random Variable: Formula for Mean of Probability Distribution

    • μ = Σ[X * P(X)]

    Example 4: Mean of Number of Heads in Three Coin Toss

    • μ = 1.5

    Quantitative Measures of Random Variable: Rounding Rule

    • Round the mean, variance and standard deviation to one more decimal place than the outcome X.
    • Reduce fractions to lowest terms.

    Example 5: Mean of Number of Spots on a Die

    • μ = 3.5

    Example 6: Mean Number of Girls in Five Children

    • μ = 2.5

    Example 7: Mean Number of Trips Lasting Five Nights or More

    • μ = 1.23

    Quantitative Measures of Random Variable: Variance and Standard Deviation

    • Variance of a probability distribution (σ²): σ² = Σ[(X² * P(X)] - μ²
    • Standard Deviation of a probability distribution (σ): σ = √σ²

    Example 7: Variance and Standard Deviation

    • Calculation of variance (σ²) and standard deviation (σ) from example 5.
    • σ² = 2.917
    • σ = 1.7079

    Quantitative Measures of Random Variable: Expectation

    • Expected value (or expectation) of a discrete random variable in a probability distribution is the theoretical mean value.
    • μ = E(X) = Σ[X * P(X)]

    Example 8: Expected Value of Gain

    • $0.65

    Discrete Random Variable: Binomial Random Variable

    • A binomial variable describes the number of successes in a fixed number of independent trials.
    • Each trial has only two outcomes: success or failure.
    • The probability of success (p) is constant for each trial.

    Binomial Random Variable: Mean and Variance

    • Mean: μ = np
    • Variance: σ² = npq (where q = 1 - p)

    Example 9: Probability of at Least One Hit in Three Official Times at Bat

    • Probability of at least one hit is 0.692

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    EMAT0503 Discrete Probability Distribution PDF

    Description

    Test your understanding of discrete probability distributions and random variables. Explore concepts such as probability calculations, sample spaces, and real-life examples like flipping coins and rolling dice. This quiz will help solidify your grasp of these fundamental statistics concepts.

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