Sample Spaces, Probabilities, and Unusual Events

Probability Experiments and Sample Spaces

Probability Experiment

  • A probability experiment is one where the individual outcome is unknown, but the long-run behavior is predictable.
  • Example: Tossing a fair coin. We don't know the outcome of a single toss, but over many tosses, we expect about half heads and half tails.

Probability of an Event

  • The probability of an event is the proportion of times the event occurs in the long run.
  • For a fair coin:
    • P(Heads) = \frac{1}{2}
    • P(Tails) = \frac{1}{2}

Law of Large Numbers

  • As a probability experiment is repeated, the proportion of times a given event occurs approaches its probability.

Sample Space

  • The collection of all possible outcomes of a probability experiment is called a sample space.
  • Examples:
    • Toss of a coin: Sample space = {Heads, Tails}
    • Roll of a die: Sample space = {1, 2, 3, 4, 5, 6}
    • Selecting a student at random from a list of 10,000: Sample space = {10,000 students}

Probability Model and Events

  • An event is a collection of outcomes from the sample space.
  • Example: Rolling an odd number on a die corresponds to the event {1, 3, 5} from the sample space {1, 2, 3, 4, 5, 6}.
  • A probability model specifies the probability of each event.

Notation

  • P(A) denotes the probability of event A.
  • The probability of an event is always between 0 and 1, inclusive: 0 \leq P(A) \leq 1
  • If A cannot occur, P(A) = 0
  • If A is certain to occur, P(A) = 1

Probabilities with Equally Likely Outcomes

  • If a sample space has N equally likely outcomes and an event A has k outcomes, then:
    • P(A) = \frac{\text{Number of outcomes in A}}{\text{Number of outcomes in the sample space}} = \frac{k}{N}
  • Example: Georgia Cash Four Lottery (numbers 0-9999)
    • What is the probability that all four digits are the same?
    • There are 10,000 equally likely outcomes.
    • There are 10 outcomes where all digits are the same: 0000, 1111, 2222, …, 9999.
    • P(\text{all four digits are the same}) = \frac{10}{10,000} = 0.001

Sampling from a Population

  • Sampling an individual from a population is a probability experiment.
  • The population is the sample space, and members are equally likely outcomes.
  • Example: 10,000 families in a town.
    • 4,753 own a house
    • 912 rent an apartment
    • 2,857 rent a house
    • 1,478 own a condo
    • P(\text{family owns a house}) = \frac{4,753}{10,000} = 0.4753
    • Number of families who rent = 912 + 2,857 = 3,769
    • P(\text{family rents}) = \frac{3,769}{10,000} = 0.3769

Unusual Events

  • An unusual event is one that is not likely to happen; it has a small probability.
  • Rule of thumb: Any event with P(A) < 0.05 is considered unusual.
  • Example: College with 5,000 students, 150 are math majors.
    • A randomly selected student is a math major. Is this unusual?
    • P(\text{student is a math major}) = \frac{150}{5,000} = 0.03
    • Since 0.03 < 0.05, this is considered an unusual event.

Empirical Method to Approximate Probability

  • Use observed data to approximate probabilities.
  • Example: Centers for Disease Control data on births.
    • 313,752 low birth weight babies (< 2500 grams)
    • 3,477,960 babies with weight > 2500 grams
    • Approximate the probability that a newborn baby is low birth weight.
    • Total births = 3,477,960 + 313,752 = 3,791,712
    • P(\text{low birth weight}) = \frac{313,752}{3,791,712} = 0.0827

Coin Toss Example

  • A fair coin is tossed three times. H = heads, T = tails.

    • List all eight outcomes in the sample space.
    • Assuming outcomes are equally likely, find the probability that all three tosses are heads.
    • Assuming outcomes are equally likely, find the probability that the tosses are all the same.
    • Assuming outcomes are equally likely, find the probability that exactly one of the three tosses is heads.

  • Sample Space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

  • If a sample space has n equally likely outcomes and event A has k outcomes:

    • P(A) = \frac{k}{n}

    • P(\text{all three tosses are heads}) = \frac{1}{8}

    • P(\text{tosses are all the same}) = \frac{2}{8} = \frac{1}{4}

    • P(\text{exactly one toss is heads}) = \frac{3}{8}