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:

  • $P(A) = \frac{k}{n}$

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

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

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