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Questions and Answers
What does probability measure?
What does probability measure?
Probability measures the likelihood of an event occurring in the future.
What is the probability of an event that cannot occur?
What is the probability of an event that cannot occur?
How can we express probability?
How can we express probability?
Probability can be expressed as a fraction, decimal, or percent.
A probability experiment is always conducted in a laboratory setting.
A probability experiment is always conducted in a laboratory setting.
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What is the set of all possible outcomes of a probability experiment called?
What is the set of all possible outcomes of a probability experiment called?
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What are the individual outcomes in the sample space called?
What are the individual outcomes in the sample space called?
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A ______ event is any event that can be decomposed into other events.
A ______ event is any event that can be decomposed into other events.
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A ______ event cannot be decomposed.
A ______ event cannot be decomposed.
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What is the formula for calculating the probability of an event?
What is the formula for calculating the probability of an event?
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Define an event in probability.
Define an event in probability.
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Study Notes
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
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Description
This quiz covers the fundamental principles of probability, including definitions and the concept of probability experiments. It explores how probability quantifies the likelihood of events and provides real-life examples of probability experiments. Test your understanding of these key concepts!
