Introduction to Probability

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Introduction to Probability
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What is it?

• For a random phenomenon, individual outcomes are
  not certain, but there is a regular distribution
  of outcomes in the long run
• The probability of an outcome is its long-term
  relative frequency.

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Probability vs. inference

• If a coin is tossed 100 times, you would expect
  to see 50 heads
• However, there is some (non-zero) probability
  that it only comes up heads 25 times
• If there were 100 flips, and it only came up
  heads 25 times, you might infer that it wasnt a
  fair coin.

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What are the possible outcomes?

• Want to make a list of possible outcomes, then
  find probability for each outcome
• Sample space is the set of all possible outcomes
• Events are specific outcomes or set of outcomes
  in the sample space

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Probability Scale
You can draw a scale from Impossible to Certain
What is the probability of scoring less than 5
when rolling a fair 6-sided die?
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Introduction to Probability

• Rather than using a descriptive scale, we use a
  numerical scale
• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a
  scale from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is
  almost certain to occur.

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Useful facts about probability

• Probability cannot be less than 0 or greater than
  1.
• All possible outcomes together must have
  probability 1.
• Probability of an event occurring is 1 minus
  probability that it does not occur.
• If 2 events have no outcomes in common,
  probability that one or the other occurs is sum
  of their individual probabilities

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Example You roll a six-sided die whose sides
are numbered from 1 through 6. Find the
probability of a. rolling a 4 b. rolling an
odd number c. rolling a number less than 7
number of ways to roll a 4 number of
ways to roll the die

• 1
• 6

number of ways to roll an odd number
number of ways to roll the die
3 6
1 2
number of ways to roll less than 7
number of ways to roll the die
6 6
1
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Combining Probabilities

• What is the probability of scoring either a 2 or
  a 6 from a single roll of a fair die?
• Number of ways an event can happenTotal number
  of possible outcomes

So p(1 or 5)
And p(5)

Or p(1)

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Combining probabilities

• What happens if we need to calculate the
  probability of one event occurring and another
  event occurring?
• What is the probability of rolling a 4 with a die
  and tossing a head with a coin?
• How many outcomes are there?

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Combining probabilities
12 possible outcomes.
Only one is H plus a 4
So p(4 and head)
Or The probability of rolling a 4 is
The probability of throwing a head is


of
So p(4 and head)
of the time
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Probability

• Throwing dice, tossing coins, picking cards - in
  each case each outcome is equally likely
• It is not always the case that all possible
  outcomes of a trial are equally likely.
• Sometimes we cannot calculate the probability,
  but can only estimate it

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Probability

• Will it rain tomorrow? Will Team A win over Team
  B?
• Each outcome is not equally likely.
• For one team playing a game against another,
  there are three possible outcomes Win, Lose,
  Draw
• Probability of winning is not one third.

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Probability

• The probability that they will win can only be
  estimated, using other information, mainly past
  performance of the two teams.