AP Statistics Curriculum 2007 Prob Basics

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General Advance-Placement (AP) Statistics Curriculum - Fundamentals of Probability Theory

Fundamentals of Probability Theory

Probability theory plays role in all studies of natural processes across scientific disciplines. The need for a theoretical probabilistic foundation is obvious since natural variation effects all measurements, observations and findings about different phenomena. Probability theory provides the basic techniques for statistical inference.

Random Sampling

A simple random sample of n items is a sample in which every member of the population has an equal chance of being selected and the members of the sample are chosen independently.

  • Example: Consider a class of students as the population under study. If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, it is the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the Axiom of Choice.
  • Questions:
    • How would you go about randomly selecting five students from a class of 100?
    • How representative of the population is the sample likely to be? The sample won’t exactly resemble the population as there will be some chance variation. This discrepancy is called chance error due to sampling.
  • Definition: Sampling bias is non-randomness that refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.

Hands-on activities

  • Monty Hall (Three-Door) Problem: Go to SOCR Games and select the Monty Hall Game. Click in the Information button to get the instructions on using the applet. Run the game 10 times with one of two strategies:
    • Stay-home strategy - choose one card first, as the computer reveals one of the donkey cards, you always stay with the card you originally chose.
    • Swap strategy - choose one card first, as the computer reveals one of the donkey cards, you always swap your original guess and go with the third face-down card!
  • You can try the Monty Hall Experiment, as well. There you can run a very large number of trials automatically and empirically observe the outcomes. Notice that your chance to win doubles if you use the swap-strategy. Why is that?

Law of Large Numbers

When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards the relative frequencies, not absolute counts of heads and tails.

  • There are two widely held misconceptions about what the law of large numbers about coin tosses:
    • Differences between the actual numbers of heads and tails become more variable with increase of the number of tosses – a sequence of 10 heads doesn’t increase the chances of selecting a tail on the next trial.
    • Coin toss results are independent and fair, and the outcome behavior is unpredictable.

Types of probabilities

Probability models have two essential components: sample space and probabilities.

  • Sample space (S) for a random experiment is the set of all possible outcomes of the experiment.
    • An event is a collection of outcomes.
    • An event occurs if any outcome making up that event occurs.
  • Probabilities for each event in the sample space.

Where do the outcomes and the probabilities come from?

  • Probabilities may come from models – say mathematical/physical description of the sample space and the chance of each event. Construct a fair die tossing game.
  • Probabilities may be derived from data – data observations determine our probability distribution. Say we toss a coin 100 times and the observed Head/Tail counts are used as probabilities.
  • Subjective Probabilities – combining data and psychological factors to design a reasonable probability table (e.g., gambling, stock market).

Event Manipulations

Just like we develop rules for numeric arithmetic, we like to use certain event-manipulation rules (event-arithmetic).

  • Complement: The complement of an event A, denoted Ac or A', occurs if and only if A does not occur.
  • A \cup B, read "A or B", contains all outcomes in A or B (or both).
  • A \cap B, read "A and B", contains all outcomes which are in both A and B.
  • Draw Venn diagram pictures of these composite events.
  • Mutually exclusive events cannot occur at the same time (A \cap B=0).

Axioms of probability

  • First axiom: The probability of an event is a non-negative real number: P(E)\geq 0, \forall E\subseteq  S, where S is the sample space.
  • Second axiom: This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space: P(S) = 1. This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.
  • Third axiom: This is the assumption of additivity: Any countable sequence of pair-wise disjoint events E1,E2,... satisfies P(E_1 \cup E_2 \cup \cdots) = \sum_i P(E_i).
  • Note: For a finite sample space, a sequence of number { p_1, p_2, p_3, \cdots, p_n} is a probability distribution for a sample space S = {s_1, s_2, s_3, \cdots, s_n}, if the probability of the outcome sk, p(sk) = pk, for each 1 \leq k \leq n, all p_k \geq 0 and \sum_{k=1}^n{p_k}=1.

Birthday Paradox

The Birthday Paradox Experiment provides an interesting illustration of some of the fundamental probability concepts.

In a random group of N people, what is the probability, P, that at least two people have the same birthday?

  • Example, if N=23, P>0.5. Main confusion arises from the fact that in real life we rarely meet people having the same birthday as us, and we meet more than 23 people.

The reason for such high probability is that any of the 23 people can compare their birthday with any other one, not just you comparing your birthday to anybody else’s.

There are N-Choose-2 = 20*19/2 ways to select a pair or people from a group of 20. Assume there are 365 days in a year, P(one-particular-pair-same-B-day)=1/365, and P(one-particular-pair-failure)=1-1/365 ~ 0.99726.

For N=20, 20-Choose-2 = 190. E={No 2 people have the same birthday is the event all 190 pairs fail (have different birthdays)}, then P(E) = P(failure)190 = 0.99726190 = 0.59. Hence, P(at-least-one-success)=1-0.59=0.41, quite high. Note: for N=42, P>0.9.


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