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 (D)</p> Signup and view all the answers

How many outcomes are possible when flipping three coins?

<p>8 (D)</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 (C)</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 (A)</p> Signup and view all the answers

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

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

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

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

What is the formula for calculating the population mean?

<p>$ rac{ΣX}{N}$ (A)</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 (B)</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 (C)</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 (A)</p> Signup and view all the answers

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

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

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

<p>1.23 (B)</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 (B)</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 (B)</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 (D)</p> Signup and view all the answers

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

<p>2.91 (A)</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 (D)</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 (C)</p> Signup and view all the answers

How can expectation be applied in real-world scenarios?

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

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

<p>32 (B)</p> Signup and view all the answers

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

<p>10/32 (D)</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 (A)</p> Signup and view all the answers

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

<p>5 (D)</p> Signup and view all the answers

What probability corresponds to having 4 girls in the outcomes?

<p>5/32 (D)</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 (C)</p> Signup and view all the answers

Which probability corresponds to having exactly 3 boys?

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

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

<p>10 (C)</p> Signup and view all the answers

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

<p>3.5 (A)</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 (A)</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 (A)</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 (C)</p> Signup and view all the answers

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

<p>21 (A)</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 (A)</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. (C)</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 (D)</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 (A)</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 (C)</p> Signup and view all the answers

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

<p>$n imes p imes q$ (C)</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 (B)</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 (A)</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. (D)</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 (A)</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 (B)</p> Signup and view all the answers

Flashcards

Random Variable

A variable whose value is determined by a random experiment.

Discrete Probability Distribution

A table that lists the probabilities for each outcome of a discrete random variable.

What are the requirements for Probability Distributions?

  1. The sum of all probabilities must equal 1. 2. Each probability must be between 0 and 1.

σ𝑃 𝑋 = 1

The sum of all probabilities must equal 1.

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0 ≤ 𝑃(𝑋) ≤ 1

Each probability must be between 0 and 1, inclusive

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Possible Outcomes

The total number of unique combinations that can occur in a random experiment.

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Probability Distribution

A table that lists the probability of each possible outcome of a random variable.

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Discrete Random Variable

A variable whose value can only take on a finite number of values (usually whole numbers).

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What is P(X) in a Probability Distribution?

P(X) represents the probability of the random variable taking on a specific value (X).

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Sum of Probabilities

The sum of all probabilities in a probability distribution must equal 1.

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Probability Range

Each probability in a probability distribution must be between 0 and 1, inclusive.

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What does a probability of 0 represent?

A probability of 0 indicates that an event is impossible.

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What does a probability of 1 represent?

A probability of 1 indicates that an event is certain.

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Mean of a Random Variable

The average value of a random variable, calculated by summing the product of each outcome and its probability.

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Outcome (x)

A possible result of a random variable, like a specific number of spots on a die.

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Probability (P(X))

The chance of a specific outcome occurring.

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Summation (𝜮)

Represents the sum of all values obtained by multiplying each outcome (x) by its corresponding probability (P(x)).

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Mean of Discrete Probability Distribution

The average value of a discrete random variable, calculated by multiplying each outcome with its probability and summing the results.

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How to calculate Mean of Discrete Probability Distribution

The formula is µ = Σ(x * P(x)) where x represents the outcome and P(x) is the probability of that outcome.

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What is the mean number of girls in families with 5 children?

In a family with 5 children, the average number of girls is calculated by considering all possible family combinations and their probabilities, and then summing the products of the number of girls and the probabilities.

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Mean is a measure of...

The mean is a measure of central tendency, providing a representative value that summarizes the entire data set, in this case, the average number of girls in a family.

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Variance of Discrete Probability Distribution

Measures the spread of a discrete probability distribution. It calculates the average squared deviation of each outcome from the mean.

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Possible Outcomes for 5 children

The possible outcomes for the gender of five children can be visualized using a combination of 'M' (male) and 'F' (female), like MMMMF or FMFFM.

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Formula for Variance

σ² = Σ[(x² * P(x)) - µ²]

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Expected Value

The average outcome of a random variable, calculated as the weighted sum of all possible outcomes, weighted by their probabilities.

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Probability of each family combination

The probability of each specific family combination, like MMMMF or FMFFM, depends on the equal chance of having a boy or a girl for each offspring.

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Standard Deviation of Discrete Probability Distribution

The square root of the variance. It measures the average deviation of each outcome from the mean, in the same units as the original data.

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How to calculate Standard Deviation

σ = √σ² or σ = Σ[(x² * P(x)) - µ²]

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Binomial Experiment

A sequence of independent trials with only two possible outcomes (success or failure) and a constant probability of success for each trial.

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Binomial Random Variable

A variable that counts the number of successes in a binomial experiment.

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Expected Value

The theoretical average value of a discrete random variable, equivalent to the mean of the probability distribution.

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Formula for Expected Value

E(X) = µ = Σ(x * P(x))

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What is 'p' in a binomial experiment?

'p' represents the probability of success in a single trial of the binomial experiment.

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What is 'q' in a binomial experiment?

'q' represents the probability of failure in a single trial of the binomial experiment, calculated as 1 minus the probability of success 'p'.

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Mean of Binomial RV

The average number of successes in a binomial experiment, calculated as the product of the number of trials 'n' and the probability of success 'p'.

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Variance of Binomial RV

A measure of how spread out the possible outcomes of a binomial random variable are, calculated as the product of the number of trials, the probability of success, and the probability of failure.

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How to calculate probability of at least one success?

Calculate the probability of each possible number of successes (from 1 to 'n'), and add them together. Alternatively, subtract the probability of getting zero successes from 1.

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Formula for Mean (Discrete Distribution)

The mean (µ) is calculated as: µ = Σ(X * P(X)), where 'X' is the outcome and 'P(X)' is its probability.

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What does Σ(X * P(X)) represent?

It represents the weighted sum of all possible outcomes of a discrete random variable, each weighted by its probability.

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Example: Toss a coin 3 times

If we toss three coins, the mean number of heads is calculated by multiplying each possible number of heads (0, 1, 2, 3) by its corresponding probability and summing the results.

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Rounding Rule for Probability Distributions

Round the mean, variance, and standard deviation to one more decimal place than the outcome 'X' of the random variable.

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Why round to one more decimal?

It's a convention to maintain consistency and avoid misrepresentation of precision in calculations based on a probability distribution.

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Non-Probability Distributions

These are sets of values that don't meet the requirements for a probability distribution, such as negative probabilities or probabilities that don't sum to 1.

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Understanding Probability Distributions

Probability distributions are essential for understanding the behavior of random variables. They allow us to calculate measures like the mean and variance, which provide valuable insights into the expected outcomes and variability of a random phenomenon.

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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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