Probability Distribution Quiz

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Questions and Answers

What is the probability that exactly one man will live through the next year?

0.0001

What is the probability that at least one man will live through the next year?

0.9999

What is the probability that fewer than two men will live through the next year?

0.0007

What is the first step in calculating the mean of the given probability distribution?

<p>Multiply each value of <code>x</code> by its corresponding probability <code>P(x)</code> (C)</p> Signup and view all the answers

Which formula is used to calculate the variance from the probability distribution?

<p>variance = Σ [(x - mean)^2 * P(x)] (C)</p> Signup and view all the answers

What aspect of the probability distribution does the standard deviation measure?

<p>The spread or variability of the distribution (B)</p> Signup and view all the answers

If the mean of the probability distribution is found to be 3.923, what will be the result when calculating the variance using the mean?

<p>The variance will reflect the average distance of outcomes from the mean (B)</p> Signup and view all the answers

What is the relationship between the expected value and the mean in this probability distribution?

<p>They are the same conceptually and numerically (D)</p> Signup and view all the answers

Flashcards

Probability Distribution

A table or function that shows the probability of each possible outcome for a variable.

Mean (Expected Value)

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

Standard Deviation

A measure of the spread or dispersion of a probability distribution.

Variance

The average of the squared differences from the mean.

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Probability

The chance or likelihood of an event occurring.

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Outcome

A possible result of an event or experiment.

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x

The random variable representing the number of men living through the year.

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P(x)

Probability of 'x' men living through the next year.

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Expected Value formula

Σ (x * P(x))

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Variance formula

Σ [(x - mean)^2 * P(x)]

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Standard Deviation calculation

Square root of the variance.

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Spreadsheet Software

Tool for calculating mean and standard deviation, faster than manual calculation.

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Calculator

Tool for arithmetic operations, including the calculations needed for standard deviation and mean.

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∑

Sigma, symbol for summation.

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Random Variable

A variable whose value is a numerical outcome of a random phenomenon.

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Probability data

Data that shows chances for various results of an event, in a suitable format.

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Numerical outcome

A calculated result that is a number; e.g. a quantity that shows a certain value.

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Random Phenomenon

An event with uncertain outcomes. Eg: Rolling a dice.

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Probability Distribution Table

A table that lists all possible outcomes of an event and their probabilities

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Calculating mean from a probability table

Involves summarizing the probabilities of each outcome, multiplied by the associated values

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Calculating standard deviation from a probability table

Requires the variance of the outcomes, calculated from the mean.

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Calculate Expected Value

The average outcome in a probability distribution. Multiply each outcome by its probability, then add up the results.

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Spread out data?

A higher standard deviation indicates that the data is more spread out around the mean. A lower standard deviation means the data points are closer to the average outcome.

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Study Notes

Probability of Exactly One Man Living Through the Year

  • This probability requires calculating the chances of one man living and the other dying, while also accounting for which man survives
  • There are two possible scenarios:
    • Man 1 lives, Man 2 dies
    • Man 1 dies, Man 2 lives
  • Each scenario needs to be calculated individually and then added together

Probability of At Least One Man Living Through the Year

  • This probability covers all scenarios where at least one man survives
  • It can be calculated directly by finding the probability of the opposite scenario happening (both men dying) and subtracting it from 1 (certainty)

Probability of Fewer Than Two Men Living Through the Year

  • This probability covers scenarios where zero or one man survives
  • This can be calculated by:
    • Finding the probability of both men dying
    • Adding it to the probability of exactly one man living (calculated previously)

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