Introduction to Probability

1. Introduction to Probability

2. What is probability? • A number between 0 and 1 (inclusive) that gives us an idea of how likely it is that an event will occur. • An event with a probability of 0 is an impossible or null event. • An event with a probability of 1 is a sure thing or a certain event. • Closer probability is to 1, more likely it is to happen.

3. Three Ways of Assigning Probabilities to Events • Personal opinion approach • Relative frequency approach • Classical approach

4. Personal Opinion Approach • “Your guess is as good as mine” approach. • Individual specifies probability based on his or her own experiences, knowledge, hunches, … • Only constraint is that probabilities must make sense.

5. Relative Frequency Approach • Observe something a large number of times. [“Toss a coin”] • Count the number of times the event of interest occurs. [“# of heads”] • Estimate probability of event by calculating the proportion of times the event occurred [“# of heads  # of tosses”]

6. Example:Relative Frequency Approach Tosser #(Tosses) #(Heads) P(H) Buffon 4,040 2,048 0.5069 Pearson 24,000 12,012 0.5005 Kerrich 10,000 5,067 0.5067 Note: To use this approach, you cannot observe process only a few times. Must be able to observe process enough to see what happens in “the long run”!

7. Classical Approach • Approach only “legit” if each outcome is equally likely. • Count total number of possible outcomes. (“52 cards”) • Count number of ways the event can occur (“4 aces”) • Divide # of event ways by # of total ways to get probability (P = 4/52 = 0.077)