Probability Distributions Quiz

Probability Distributions

• A probability distribution is a function that describes the likelihood of obtaining different possible outcomes in an experiment
• Key properties of a probability distribution:
  • The probabilities of all possible outcomes must add up to 1
  • The probabilities must be between 0 and 1

Determining Probability Distributions

• Question One a):

  • The probabilities for all the outcomes of X add up to 1, so the table represents a probability distribution.

• Question One b):

  • The probabilities for all the outcomes of X add up to 1, so the table represents a probability distribution.

Mean, Variance, and Standard Deviation

• The mean of a discrete probability distribution is the average value of the random variable, weighted by the probabilities of each value
• The variance measures the spread of the distribution. It is the average of the squared deviations from the mean
• The standard deviation is the square root of the variance, indicating the average distance from the mean.

Finding Mean, Variance, and Standard Deviation

• Question Two:
  • To find the mean: Multiply each possible outcome by its probability, and sum the results
  • To find the variance: Multiply each outcome's probability by the square of the difference between the outcome and the mean, and sum the results
  • To find the standard deviation: Take the square root of the variance

The Binomial Probability Distribution

• The binomial distribution is a probability distribution that describes the probability of obtaining a specific number of successes in a sequence of independent trials.
• The trials must have exactly two possible outcomes: success or failure.
• The likelihood of success (p) should remain constant over all the trials.

Finding The Probability of Success

• Question Three:
  • The probability of success (p) is 75%
  • The probability of failure (q) is 25%
  • There are 7 trials (n)
  • We need to find the probability of exactly 4 successes (x)
• The formula to calculate this probability is: (n choose x) * (p^x) * (q^(n-x))
  • Where (n choose x) is the binomial coefficient calculated as (n!)/ (x! * (n-x)!)