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Probabilities of Events

Probabilities of Events. You should think of the probability of an event A ⊆ S a s the weight of the subset A relative to the weight of the current universe S (it follows that the probability of S is 1 ) In this light the following formulas are kind of obvious:

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Probabilities of Events

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  1. Probabilities of Events You should think of the probability of an event A ⊆ S as the weight of the subset A relative to the weight of the current universe S (it follows that the probability of S is 1) In this light the following formulas are kind of obvious: • P(A∪B) = P(A) + P(B) – P(A∩B) • P(Ac) = 1 – P(A)

  2. Formula 2 is obvious because A and Ac add up to S. Formula 1 P(A∪B) = P(A) + P(B) – P(A∩B) is equally obvious from the figure Since the weight of A∩B is counted twice in P(A) + P(B)

  3. Formula 1 is called by our textbook The Additive Rule of Probability Other books call it The Inclusion/Exclusion Principle Whatever we call it, I prefer remembering it symmetrically as P(A∪B) + P(A∩B)= P(A) + P(B) (I don’t have to remember where the minus sign goes!) Any problem dealing with two events must give you enough information to determine, maybe using also the Rule of Complements P(Ac) = 1 – P(A)

  4. three of the four numbers P(A∪B),P(A∩B), P(A) and P(B) You compute the fourth one and fill the four spaces in the Venn diagram

  5. Important Advice When the problem deals with two events do not read the question! First fill the four spaces in the Venn diagram, Then read and answer the question(s) !

  6. If a problem deals with threeevents do not read the question! First fill the eightspaces in the Venn diagram, (The numbering of the spaces is arbitrary) Then read and answer the question(s) !

  7. Conditional Probability Recall that the probability P(A) of an event A ⊆ S can be thought as the percentage of S embodied by A There are situations when we will be interested in determining what percentage of B is embodied by A for some given event B. (instead of S) Here are a couple of simple examples: Toss a pair of fair dice. Let

  8. A = the sum is even B = the sum is 7 or less. The figure below shows that P(A) = = (the percentage of S embodied by A) =

  9. But now we ask: what percentage of B is embodied by A ? (In the language of gamblers, betting on A gives a fifty-fifty chance of winning, but should you change your bet if you are told that B happened?) The figure in the next slide shows B in celeste, with those entries of A which are part of B highlighted (larger and embossed.)

  10. what percentage of B is embodied by A ? (COUNT !)

  11. That’s right, , less than !! (change your bet!) Here is another example. The table in the next slide shows the number, type and country of manufacture of the vehicles parked in a local WalMart parking lot yesterday at noon. Let A = the vehicle is of foreign manufacture and B = the vehicle is a passenger car

  12. I assert P(A) = and Percentage of B embodied by A =

  13. The two examples suggest that: We need a symbol for percentage of B embodied by A and What has this got to do with probabilities? We have encountered the symbol already, and the reason that the notion of probability is inherent here is that, focusing entirely on the event B we are asking the question what are the chances of A ifBis our new universe?

  14. Pictorially we are changing ouruniversefrom to

  15. Therefore in terms of relative weight, the percentage of B embodied by A Is simply the weight of A∩B relative to the weight of B i.e. (remember?) As for the symbol (next slide …)

  16. the percentage of B embodied by A is written thus P(A|B) and read the probability of A given B P(A|B) = Now do all the homework assigned.

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