AP Statistics Section 6.2 A Probability Models

If you tossed a coin 5 times, would you be surprised if you got heads exactly 1 time? If you tossed a coin 50 times, would you be surprised if you got heads exactly 1 time?

The difference in your answers is explained by the idea that chance behavior is ____________ in the short run but has a __________________ in the long run. unpredictable predictable behavior

The word random in statistics is not a synonym for “haphazard” but a description of a kind of _______ that emerges only in the _________. pattern long run

We often encounter the unpredictable side of randomness in our everyday experience, but we rarely see enough repetitions of the same random phenomenon to observe the long-term regularity that probability describes. In the very long run, the proportion of heads is 0.5. This is the intuitive side of probability. A probability of 0.5 means “occurs half the time in a very large number of trials.”

We call a phenomenon random if individual outcomes are uncertain but there is, nonetheless, a regular distribution of outcomes in a large number of repetitions.

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a large number of repetitions.

In other words, probability is long term relative frequency

empirical observation The idea of probability is __________. That is, it is based on ____________ rather than theorizing. Probability describes what happens in very many trials, and we must actually observe many trials to pin down a probability. empirical observation

The sample space (S) of a random phenomenon is the set of all possible outcomes.

An event is any outcome or set of outcomes of a random phenomenon An event is any outcome or set of outcomes of a random phenomenon. An event is always a subset of the sample space.

2. A way of assigning probabilities to events. A probability model is a mathematical description of a random phenomenon consisting of two parts: 1. The sample space. 2. A way of assigning probabilities to events.

Example: What is the sample space for …. a. rolling two dice Example: What is the sample space for …. a. rolling two dice? (Think of rolling a red die and green die.)

Example: What is the sample space for …. b) tossing 4 coins?

Example: What is the sample space for …. c Example: What is the sample space for …. c. flipping a coin followed by throwing a die

Being able to determine the outcomes in a sample space is critical to determining probabilities. One way to do this is by using a ___________. The tree diagram part c above looks like: tree diagram

The previous example illustrates the Multiplication Principle: If you can do one task in n1 ways and a second task in n2 ways, then both tasks can be done in ______ number of ways.

If you are drawing playing cards twice from a standard deck of 52 cards, then the second draw depends on what you do with the first card. If you put the first card back in the deck before drawing the second card, that is sampling ________________. If you do not put the first card back in the deck before drawing the second card, you are sampling __________________. with replacement without replacement

Example: How many 3 letter “words” are there if letters can be repeated?

Example: How many 3 letter “words” are there if letters cannot be repeated?