Computing Basic Probabilities

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Computing Basic Probabilities

• Tuesday 13 November 2007

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Two Types of Probability

• True probability or frequency
• vs
• Experimental probability

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The Law of Large Numbers

• The true probability of flipping a head on a coin
  is ½.
• Of course, on a single throw, youre guaranteed
  not to get ½ of a head.
• Youll get either 1 head, or 0.
• The LoLN says that as the number of flips
  increases, the experimental frequency of heads
  will approach ½.
• Approach here is complicated. Well discuss.

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Independence

• Two events are independent if the outcome of one
  has no influence on the outcome of the other.
• Events that are not independent are said to be
  dependent.

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The Sample Space for an Event

• Identify the sample space if I roll two dice,
  add, and then flip a coin.

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• 1 H
• 2 H
• 3 H
• 4 H
• 5 H
• 6 H

• 1 T
• 2 T
• 3 T
• 4 T
• 5 T
• 6 T

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Sample Space Tree Diagram
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Simple vs. Non-Simple Outcomes

• Simple outcomes consist of a single possible
  outcome.
• For example, rolling a 6.
• Outcomes that arent simple consist of several
  possible outcomes, all of which satisfy some
  condition.
• For example, not rolling a 6.

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Theoretical Probability

• A probability is a number between 0 and 1 that
  describes the likelihood that an event will take
  place.

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Qualities of Probability

• For any event A,
• 0 P(A) 1
• If P(A) 1, A is definitely going to happen.
• If P(A) 0, A is definitely not going to happen.

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Example

• Suppose we know that one of three things is going
  to happen
• You get an A in statistics.
• You get a B in statistics.
• You get a lousy grade in statistics.
• If the probability of an A is 20 , and the
  probability of a B is 25 , what is the
  probability that you will get a lousy grade?

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The Complement
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The Complement of an Event

• The complement of an event A is the set of all
  outcomes in which A does not happen.
• The complement of A is written
• AC
• In addition, we know that
• P(AC) 1-P(A)

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The Union of Two Events

• The union of two probabilities is the
  probability that one or the other will occur.
• It is written as P(A U B).
• If the two events are mutually exclusive that
  is, if they cant both happen then
• P(A U B) P(A) P(B)

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The Union of Two Events

• These events are mutually exclusive since they
  cant both be true at once.
• If P(A) 0.8 and
• P(B) 0.05, then
• P(A U B) 0.8 0.05 0.85

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The Intersection of Two Events

• The intersection of two probabilities is the
  probability that both will occur.
• It is written as P(A ? B).
• If the two events are independent then
• P(A ? B) P(A) X P(B)

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The Intersection of Two Events

• If the two events are independent
• P(A) 0.16 and
• P(B) 0.25, then
• P(A ? B) 0.16 X 0.25 0.04.

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Computing Probabilities

• No easy system.
• READ.
• THINK.
• Is this problem solvable using addition or
  subtraction?
• Or And Neither
• Will I need to multiply?
• Both All several tries

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• .3 .3 .9 0
• .027 .512 .271
• .72 .89 .28
• .5184 .0784
• .375 .6875 .3125 .5
• On each spin, player A has only a .25 chance of
  winning.
• .06 .50 .94
• .45 .87
• .3025 .2025 .7569

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Summary

• There are two kinds of probability
• Experimental (or Empirical)
• Theoretical (or True)
• A sample space is the list of all the possible
  outcomes of a random event.
• Order generally matters in defining a sample
  space.
• Independent events
• Probabilities are added for or
• Probabilities are multiplied for and