Unit 4

• Binomial distribution


  A probability distribution for a discrete random variable representing the number of successes in N repeated trials
• Calculating binomial probabilities


  1. List all possible outcomes for X (number of successes)

  2. Calculate probability of each outcome using formula: nCr * p^r * (1-p)^(n-r)

  3. Where n=number of trials, r=number of successes, p=probability of success, (1-p)=probability of failure
• Binomial distribution


  • Requires success clearly defined and probability of success given
  • Probability of success stays the same from trial to trial
  • Trials are independent
• Geometric distribution


  Discrete random variable where X is the number of the trial on which the first success occurs
• Geometric distribution


  • Requires success clearly defined and probability of success given
  • Probability of success stays the same from trial to trial
  • Trials are independent
  • No set number of trials, only care about when first success occurs
• Mean of binomial distribution


  N * P
• Standard deviation of binomial distribution

  Square root of N * P * (1-P)
• Mean of geometric distribution

  1 / P
• Standard deviation of geometric distribution

  Square root of (1-P) / P
• Random process

  Generates results that are random or simply unknown and determined by chance
• Outcome

  The result of a random process
• Event

  The collection of outcomes
• Probability

  Quantifying the uncertainty in a random process
• Long run relative frequency

  The probability of an outcome, found by taking the number of times the outcome occurs divided by the total number of repetitions
• The law of large numbers states that simulated probabilities tend to get closer to the true probability the more trials we perform
• The long run relative frequency definition of probability will never be perfect because we can never run infinite simulations or trials
• Sample space

  A list of all non-overlapping outcomes
• Probability of an event

  The number of outcomes in favor of the event divided by the total number of outcomes in the sample space
• The probability of an event is always between 0 and 1 inclusive
• Complement of an event

  The event does not happen
• Probability of A and B

  The probability that both events A and B occur at the same time (joint probability)
• Mutually exclusive events

  Events that cannot happen at the same time
• Probability of A or B

  The probability that event A only happens, or event B only happens, or both A and B happen (union of events)
• Conditional probability

  The probability of event A occurring given or on the condition that event B has already or will occur
• Addition rule

  To find the probability of A or B, take the probability of A plus the probability of B, and subtract the probability of A and B
• Multiplication rule

  To find the probability of A and B, take the probability of A times the probability of B given that A has already occurred
• Independent events

  Events where the probability of one event is not affected by the occurrence of the other event
• If events A and B are independent, the probability of A and B is the probability of A times the probability of B
• If events A and B are not independent, the probability of A and B is the probability of A times the probability of B given that A has already occurred
• Two-way table

  A table that shows the relationship between two categorical variables
• The AP Statistics Exam loves working with two-way tables and probability
• Probability question with two-way table

  1. Identify the number of outcomes in favor

  2. Identify the total number of outcomes in the sample space

  3. Calculate the probability
• Joint probability

  The probability of two events occurring together
• Mutually exclusive

  Two events that cannot occur at the same time
• Calculating probability with mutually exclusive events

  1. Add the individual probabilities

  2. Subtract the overlap to avoid double counting
• Conditional probability

  The probability of one event occurring given that another event has occurred
• Calculating conditional probability

  1. Numerator: Probability of both events

  2. Denominator: Probability of the condition
• Independence

  Two events are independent if the occurrence of one event does not affect the probability of the other event
• Checking for independence

  1. Compare the probability of one event to the conditional probability of that event given the other event

  2. If they are equal, the events are independent
• The probability that Andrea purchases Car 1 is 42% and the probability she purchases Car 2 is 26%