Basic Probability Concepts

Basic Probability

• Probability is a loosely defined term used in everyday conversation, representing one's belief in a future event happening.
• Probability is a number between 0 and 1 (inclusive), reflecting the likelihood of an event occurring.
• Probability describes the chance of an event occurring if an activity is repeated many times.
• Probability of 0 means an event cannot happen, while a probability of 1 indicates an event must occur.

Probability Experiment

• A probability experiment is the process that yields an observation or measurement.
• Experiments, as used in this context, are controlled laboratory situations.
• More broadly, any situation involving chance outcomes are considered experiments.

Probability Experiments Examples

• Taking an exam
• Tossing a fair coin
• Rolling a fair die
• Delivering a sales pitch
• Buying a defective cellphone
• Having a preference for a certain cola brand
• Choosing a number from a set
• Examining a fuse for quality control

Sample Space

• The sample space is the set of all possible outcomes of an experiment.
• The outcomes in the sample space are called sample points.

Example: Toss Coin Twice

• The Sample space is {HH, HT, TH, TT}.

Exercise (Sample Spaces): Toss Coin 3 Times

• The total number of outcomes is 2³ = 8.
• The sample space is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.

Exercise (Sample Spaces): Rolling Two Dice

• Rolling two dice gives 36 possible outcomes.

Exercise (Sample Spaces): Record Sex of Successive Children in a Three-Child Family

• There are 8 possible outcomes.
• The sample space is {BBB, BBG, BGB, GBB, BGG, GGB, GGG, GGB}.

Exercise (Sample Spaces): Choosing a Card from a Standard Deck

• There are 52 possible outcomes
• The sample space includes all 52 cards in a standard deck (Hearts, Clubs, Spades, Diamonds).

Exercise (Sample Spaces): Taking a 10-Question Survey

• There are 1024 possible outcomes when a survey with two choices (yes or no) is taken for each question.

Events

• Events are sets of outcomes from a sample space.
• An event, denoted by E, is a subset of the sample space (E ⊆ S).
• An event happens if the experiment's outcome is in the event set.
• Events are classified as simple or compound.

Example: Tossing a Die

• Let S = {1, 2, 3, 4, 5, 6}
• Possible events:
  • E₁: Getting a one (E₁ = {1})
  • E₂: Observing a five (E₂ = {5})
  • E₃: Getting an odd number (E₃ = {1, 3, 5})
  • E₄: Observing a number greater than 4 (E₄ = {5, 6})
  • E₅: Not observing a two (E₅ = {1, 3, 4, 5, 6})

Simple vs. Compound Events

• A compound event can be broken down into simpler events.
• Simple events cannot be further decomposed.

Probability of an Event

• If an experiment has k equally likely outcomes, and an event E contains m sample points, then P(E) = m / k = n(E) / n(S) where
  • m is the number of sample points in E
  • k is the number of sample points in the sample space S
  • n(E) is the number of outcomes in event E
  • n(S) is the number of outcomes in the sample space

Example: Probability of Events (Tossing a Die)

• Consider a fair die toss:
  • a. Finding the probability of an even number happening (E = {2, 4, 6}) leads to a probability of 1/2 = 3 outcomes/6 outcomes
  • b. Finding the probability of a number divisible by 3 happening (E = {3, 6}) leads to a probability of 1/3 = 2 outcomes/ 6 outcomes

Example 3: Probability of Events (Coin Tossed 3 Times)

• Let S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
  • E₁: Observe at least two heads. P(E₁) = 4/8 = 1/2
  • E₂: Observe exactly two heads. P(E₂) = 3/8
  • E₃: Observe at most two heads. P(E₃) = 7/8