CS 3341 Lecture Notes - Lecture 1: Pierre De Fermat, Abraham De Moivre, Blaise Pascal
16 May 2018
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PROB & STAT for COMP SCI 1. PROBABILITY BASICS Ā© Whalen
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āWe balance probabilities and choose the most likely. It is the scientific use of the imagination.ā
āSherlock Holmes, The Hound of the Baskervilles
What are the Chances?
We talk about chance all the time. What are the chances of getting a job? Of winning a game? Of
meeting someone? Of rain tomorrow? Of getting to work on time?
For day-to-day purposes, we may only need an intuitive understanding of chance, or equivalently,
probability. To use probability in math and science, however, we need a rigorous definition. This turns
out to be a highly non-trivial task, and there are multiple theories that mathematicians and scientists
have developed.
Three interpretations of probability
First interpretation:
Probabilityās origin is in the context of gambling games, particularly in France in the 1600s. Some of the
early masters include Abraham de Moivre, Blaise Pascal, Pierre de Fermat, and Pierre Simon, marquis
de Laplace.
Probability rules were developed and explained using problems involving dice, cards, spinners, coin
tosses, and random draws from a bucket (or urn). In these problems, probability is computed by
reducing events to equally-likely outcomes.
It is well-suited for game theory and events with some kind of symmetry. It is powerless to assign
probabilities to other events, such as the probability that there was ever life on Mars or that you will
meet a famous person this year, or even if it will rain tomorrow.
Second interpretation:
Extending the classical interpretation for use in statistics, probability is interpreted as the relative
frequency of an event. If we can repeat a process many times, how many of those times turned out a
certain way? For instance, if the chance of rain tomorrow is 30%, that means in our past records 30
days out of 100 like tomorrow had rain. This interpretation is fine as long as an experiment is
repeatable, but like classical probability it is powerless to assign chances to outcomes of non-
repeatable experiments.
Third interpretation:
In fact, many interesting questions cannot be answered by repeatable experiments. (Of course, an
experiment or observation may give evidence one way or another, but may not be repeatable.)
Probability may be interpreted as a degree of belief, where we assign probability values based on
evidence (and perhaps our own beliefs).
PROB & STAT for COMP SCI 1. PROBABILITY BASICS Ā© Whalen
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Each interpretation has advantages and disadvantages. Fortunately, the mathematical axioms and
rules work regardless of your interpretation.
Fun and Games
To illustrate the probability rules, we will use classical examples.
COINāa coin is flipped and lands with one of two sides facing up. One side is called āheadsā and the
other is called ātails.ā Usually a coin is assumed to be āfair,ā meaning heads and tails are equally likely.
DIE (plural: DICE)āa cube with six faces, labelled 1, 2, 3, 4, 5, and 6. Usually a die is assumed to be fair.
CARDSāa set of cards is a ādeck.ā For this class we will assume a standard poker deck. A deck of cards
has 4 suits: hearts, clubs, diamonds, and spades. There are 13 cards in each suit: 2 through 10, jack,
queen, king, and ace. So there are ī¶ īµ ī³īµ īµī·ī“ cards. Usually a deck is assumed to be āwell-shuffled,ā
implying if we draw a card from the top of the deck, each card is equally likely to be there.
Range
Mathematically, probability is a function that maps an event to a number between 0 and 1
(equivalently, a percentage between 0% and 100%).
An event is an outcome, or set of outcomes, of some experiment. āExperimentā is a generic term. It
could mean something as simple as rolling a die or drawing a card.
If an event is impossible, its probability is 0. If an event is certain, its probability is 1. To assign a
probability value to an event, we can use the classical formula:
The probability of an event is
īīīīīīīīīīīīīīīīīīīīīīīīīīī
īīīīīīīīīīīīīīīīīīīīīīīīīīīī
īīīīīīīīīīīīīīīīīīīīīīīīīīīīīīīīī
īīīīīīīīīīīīīīīīīīīīīī
assuming that each outcome is equally likely.
Multiplication Rule and Independence
Suppose you toss a coin twice. If the first toss lands heads, does that affect the second toss? If the first
toss lands tails, does that affect the second toss? No: the outcome of the first toss has no impact on
the outcome of the second toss.
Two events are independent if one of them happening has no effect on the
chance of the other happening.
We can generalize this to any list of events. Saying events are independent means that any one of
them happening makes no difference to the chances of any of the others happening. (In other words,
any two of them are independent.)
PROB & STAT for COMP SCI 1. PROBABILITY BASICS Ā© Whalen
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The probability that two events both happen uses multiplication.
To find the probability that two events both happen, first decide if they are independent.
If they are, multiply their probabilities.
Examples:
(a) A coin is tossed twice. What is the probability that it lands heads both times?
(b) A die is rolled twice. What is the chance it lands on five, then on two?
(c) A coin is tossed three times. What is the probability that it lands heads, then tails, then heads?
(d) A coin is tossed ī times. Find a formula for the probability that it lands heads on all ī tosses.
(e) Roll a die and draw a card from a well-shuffled deck. What is the probability you roll a 3 and draw
the queen of diamonds?
(f) A deck is well-shuffled. Draw the top two cards. We want the probability that the first card is the ace
of spades and the second card is the king of spades. The probability of getting the ace of spades is
1/52, and the probability of getting the king of spades is 1/52. We want the probability of both.
True or false: The probability is ī¬µ
ī¬¹ī¬¶ īµī¬µ
ī¬¹ī¬¶
