PROBABILITY Uses of Probability Reasoning about Probability Three Probability Rules The Binomial Distribution

Uses of Probability basis of inferential statistics useful in everyday life – how safe is this? – how big of a gamble is this? – is this event meaningful?

Reasoning About Probability Linda is 31 years old, majored in philosophy, and is outspoken about political issues. Which is more likely? Linda is – A. a bank teller – B. a feminist bank teller

Reasoning About Probability Assume that 5% of a population is infected with HIV, and the test for HIV has a 10% false positive rate. Assume that 100% who have HIV will test positive. For a person in this population who tests positive, what is the probability of actually having HIV?

The Achilles Heel of Human Cognition Compared to our other abilities, we are remarkably poor at reasoning about probability. We tend to use heuristics (short cuts) and common sense.

Outcome Space The list of all possible events that can occur in a particular situation An accurate listing allows accurate calculations of probability

Probability Rule 1 When each event is equally likely:

An Example What is the probability of drawing an ace out of a 52 card deck? – outcome space = {2,2,2,2,3,3,3,3….} – #chances = 4 – #possible outcomes = 52 – p(Ace) = 4/52 = 1/13 =.08 = 8%

Another Example What is the probability of getting a an odd number when rolling a fair 6 sided die? – outcome space = {1,2,3,4,5,6} – #chances = 3 – #possible outcomes = 6 – p(odd) = 3/6 =.50 = 50%

Probability Rule 2 Any probability must be between 0 and 1, or 0% and 100%. A probability of 0 means the event does not occur in the outcome space. A probability of 1 means that only that event occurs in the outcome space.

Probability Rule 3 The total probability of all events in a given situation must add up to 1 or 100%.

The Binomial Distribution Shortcut for finding probability Can be used only when: – each trial has two possible outcomes – the probability of each outcome is constant across trials – the trials are independent of each other

The Binomial Expansion Pc = probability of a combination of events N = number of trials r = number of successes p = probability of success on one trial q = 1-p

! Means factorial: Multiply the number by all whole numbers less to it down to 1 By definition, 0! = 1 Example: if N = 4 trials, then N! = 4! = 4x3x2x1 = 24

Binomial Example A number from 1 to 10 is selected at random, and you try to guess what it is. You do this 5 times. What is the probability of correctly guessing (just by chance) 3 or more times?

STEP 1: Determine p. There are 10 numbers to choose from, so the probability of guessing correctly on one trial is 1/10 or p =.10

STEP 2: Determine N and r. There are 5 trials, so N = 5. We need the probability of getting 3 or more correct, so we need to do the expansion for r = 3, r = 4, and r = 5.

STEP 3: Calculate Pc

So, P(3,4, or 5 correct) = =.00856