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Probability

Probability. Sets First we will revise sets. Sets First we will revise sets. A u B: Called “A union B” It means all of A and all of B together. A ∩ B: Called “A intersection B” It means the bit where they overlap The elements they have in common. Sets / Probability. A ∩ B :

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Probability

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  1. Probability

  2. SetsFirst we will revise sets

  3. SetsFirst we will revise sets • A u B: • Called “A union B” • It means all of A and all of B together • A ∩ B: • Called “A intersection B” • It means the bit where they overlap • The elements they have in common

  4. Sets / Probability • A ∩ B: • Called “A intersection B” • Like an intersection of two streets AND BOTH • A U B: • Called “A union B” • The total, A and B together EITHER OR

  5. Sets / Probability • A only: • Called A only • Also called A not B • Also call A less B A only • (A U B)’: • Everything except A or B • Called (A U B) compliment NEITHER

  6. Mutually Exclusive

  7. Mutually Exclusive • They can NOT happen at the same time!!! • The two events have nothing in common • No overlap in the diagram A ∩ B = 0 for mutually exclusive events

  8. MUTUALLY EXCLUSIVE Example 1 A Dice: So the sample space is: 1,2,3,4,5,6 Event B: if an it lands on ODD Event A: if an it lands on EVEN 2 4 6 1 3 5 • No numbers in common • No overlap in the diagram • MUTUALLY EXCLUSIVE • They can NOT happen at the same time!!! 1 3 5 2 4 6 MUTUALLY EXCLUSIVE

  9. NOT MUTUALLY EXCLUSIVE Example 1 A Dice: So the sample space is: 1,2,3,4,5,6 Event B: if an it lands on Prime Event A: if an it lands on EVEN 2 4 6 2 3 5 • Number in common • There IS an overlap in the diagram • NOT MUTUALLY EXCLUSIVE • These events CAN happen at the same time!!! 4 5 6 3 2 NOT MUTUALLY EXCLUSIVE

  10. Conditional Probability

  11. Conditional Probability P(A I B ) = P(A ∩ B) P(B) • What does it mean??? • Probability of A given B • Given that it is B, what is the probability of A What is the Probability of Art given that they do business Art Business 5 6 2 P(A I B ) = = 2 7 P(A ∩ B) P(B)

  12. Independent events

  13. Independent events • Two events are independent if…… the outcome of one event does not effect the outcome of the other Check for independence: P(A) x P(B) = P(A∩ B)

  14. Independent events Check for independence: P(A) x P(B) = P(A∩ B) Where does the above formula come from?? Think back to our formula for conditional probability P(A I B ) = For independence we want P (A I B) to be equal to the P(A). In other words we don’t want B to have an affect on the outcome of A So we want P (A I B) = P(A) → = P(A) Cross multiply gives: P(A) x P(B) = P(A ∩ B) P(A ∩ B) P(B) P(A ∩ B) P(B)

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