Download B i P b bilit Basic Probability Theory

Basic
B
i P
Probability
b bilit
Theory
Sahar Ahmadzadeh
University of Essex
[email protected]
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EVENTS AND PROBABILITIES
• Probability theory is a branch of mathematics that allows
us to reason about events that are inherently random.
• perform an action that can produce one of “n” different possible
random OUTCOMES, each of which is equally likely.
Then, the probability of each of those outcomes is P(A)=1/n.
• For example
• I flip a fair coin to produce one of OUTCOMES two outcomes:
P(A)=1/2, for heads or tails
heads or tails
• I pick one of the 52 different cards from a deck of playing cards at
P(A)=1/52, for each of the possible cards
random
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EVENTS AND PROBABILITIES
:event space (sample space)
Example 1: Coin
Trial: flipping a coin
Two possible outcomes: heads or tails,
p(H) is the probability of heads
if p(H) = 0:8, we would expect that flipping 100 times
would yield 80 heads
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Example 2: Die
Trial: rolling a die
outcomes: 1, 2, 3, 4, 5, or 6
event: set of results
e.g. 1 or 2 (1,2)
e.g. even (2,4,6)
e.g. distinct events: 26 = 64
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REPEATABLE EXPERIMENTS
• Flipping a coin or choosing a card from a deck at random are
both repeatcal
p
experiments.
p
• Example 3:
• Trial: flipping three coins
• Still two possible outcomes: heads or tails
• e.g. first=H, second=T, third=T (HTT)
• event: set of results
e.g. two tails and one head (A = HTT, THT, TTH)
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PROBABILITY FUNCTIONS
• 0 ((impossible)
p
) to 1 ((certain))
• p distributions probability mass 1 over the sample space
•
:function mapping sets of events to [0; 1], the
probability X
-How likely is an event to occur
-e.g. Fair
F i coin:
i
-e.g. Die: Event B = divisible by 3:
•
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AXIOMS OF PROBABILITY
• P(A) = The probability of event A.
A
• Axioms:
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PROPERTIES
• If A and B are disjoint events (sets of outcomes), i.e.
-e.g. A=roll a 3, B=roll a 6:
-e.g. A=raining, B=snowing: p(raining OR snowing) =
p(raining) + p(snowing)
•
*
notice that this subsumes the previous because we assumed
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SOME USEFUL THEOREMS
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CONDITIONAL PROBABILITY
• probability of event A given event B:
• p
prior probabiliy
p
y of A:
• posterior probability of A given B:
•
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CONDITIONAL PROBABILITY
•
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CONDITIONAL PROBABILITY
Example:
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INDEPENDENCE
If A and B are independent
p
events,, then
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INDEPENDENCE
• Example
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THE CHAIN RULE
•
•
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