Statistics Lecture 4.2: Introduction to Probability

This paragraph introduces the concept of probability as a transition from descriptive to inferential statistics. It explains the basics of probability, stating that low probability events are rare and uses coin flipping as an example. The paragraph also introduces key vocabulary such as 'event', 'simple event', and 'sample space', setting the stage for further discussion on these terms.

The speaker elaborates on the concepts of events and simple events using the example of flipping a coin. An event is a specific outcome that one is looking for, while a simple event is a single outcome from a procedure. The sample space is the set of all possible outcomes. The discussion includes the idea of listing all simple events for flipping a coin three times and understanding the relationship between events, simple events, and the sample space.

This section delves into the calculation of simple and total events, particularly focusing on the number of ways an event can be accomplished. It uses the example of flipping a coin three times and finding the number of heads and tails. The speaker emphasizes understanding the difference between an event (what is being looked for) and simple events (the ways to accomplish that event), and how they relate to the sample space.

The speaker clarifies the relationship between procedures, events, simple events, and sample spaces. A procedure is the action being performed, an event is a specific outcome of that procedure, a simple event is an individual outcome, and the sample space is the collection of all possible outcomes. The discussion includes examples of flipping a coin and rolling a die to illustrate these concepts and how they interrelate.

This paragraph introduces the three types of probability: observed, classical, and subjective. Observed probability is based on actual data from performed experiments, classical probability is based on theoretical equal chances of outcomes, and subjective probability is an estimate based on educated guesses or personal judgment. The speaker provides examples from everyday life, such as batting averages in baseball and doctors' estimates of health outcomes, to illustrate these types of probability.

The speaker calculates the classical probability of selecting a heart from a standard deck of cards. By dividing the number of hearts (13) by the total number of cards (52), the probability is found to be 0.25 or 25%. This example demonstrates how to calculate the probability of a simple event in a classical sense, assuming each card has an equal chance of being selected.

Using historical data, the speaker calculates the probability of Peyton Manning completing a pass. By dividing the number of completed passes (385) by the total number of passes attempted (528), the probability is approximately 72.9%. This example illustrates the calculation of observed probability, as it is based on actual performance data.

The speaker calculates the classical probability of randomly selecting a two from a standard deck of cards. There are four twos in the deck, so the probability is 4 out of 52, or approximately 7.7%. This example shows how to calculate the probability of a simple event without conducting an actual experiment, based on the theoretical distribution of cards.

The speaker discusses a poll on public opinion about cloning, where 91 out of 901 respondents viewed cloning positively. This is an example of observed probability, as it is based on actual data collected from a poll. The calculated probability is approximately 10.1%, indicating the likelihood of a randomly selected person from the poll sample having a positive view on cloning.

The speaker introduces the concept of subjective probability, which is based on personal judgment rather than data or theory. An example is given where the speaker humorously estimates the probability of a bird pooping on a car. The speaker also introduces the idea of complementary events, which are mutually exclusive and together encompass all possible outcomes, such as rolling a five or not rolling a five on a die.

The speaker calculates the classical probability of having two boys if a couple has three children, assuming equal chances of having a boy or a girl. The sample space is listed with all possible gender combinations, and the probability is found by dividing the number of favorable outcomes (three ways to have two boys and one girl) by the total number of outcomes (eight total combinations). The probability is 37.5%, illustrating the calculation of classical probability based on theoretical chances.

The speaker discusses the laws of probability, emphasizing that probabilities always fall between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The law of large numbers is introduced, stating that as the number of trials increases, observed probability will approach classical probability. The speaker also explains the concept of complementary events and their probabilities summing up to 1, as they cover all possible outcomes.