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STA107 Lectures 6 & 7 Conditional Probability

P(A|B) The probability our outcome is one of the simple events in A, given that we know it is one of the simple events in B OR We know our experimental outcome is one of the simple events in B. What is the probability it is also a simple event in A?

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STA107 Lectures 6 & 7 Conditional Probability

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  1. P(A|B) • The probability our outcome is one of the simple events in A, given that we know it is one of the simple events in B • OR • We know our experimental outcome is one of the simple events in B. What is the probability it is also a simple event in A? • provided P(B) ≠ 0 STA107 Lectures 6 & 7Conditional Probability

  2. Example Toss three fair coins, and look at upper faces. What is the probability at least 3 heads appear given at least 1 head appeared?

  3. Example Toss a die and look at number of spots on upper face. Let A be the event even # spots Let B be the event prime # spots P(A|B) = ? P(B|A) = ? P(A|B) = ? P(B|A) = ? P(A|B) = ? P(B|A) = ? Example

  4. P(A) = P(A|B1)P(B1) + . ... + P(A|Br)P(Br) = Law of Total Probability

  5. Example Box 1 has 5 red balls and 5 green balls, Box 2 has 7 red and 3 green, Box 3 has 6 red and 4 green. First, a box randomly is selected. The probability of choosing Box 1 is 1/2, Box 2 is 1/6, and Box 3 is 1/3. Second, a ball is randomly selected from the chosen box. What is the probability that the ball chosen is green? Box 1 Box 2 Box 3

  6. Define the following events: Let G be the event that a green ball chosen.Let B1 be the event Box 1 is selectedLet B2 be the event Box 2 is selected Let B3 be the event Box 3 is selected P(G B1) = ¼ P(G B1) = ¼ P(G B2) = 1/20 P(G B2) = 7/60 P(G) = P(G B1) + P(G B2) + P(G B3) = 13/30 P(G B3) = 2/15 P(G B3) = 1/5

  7. Example A factory has three assembly lines. 30% of items are made on Line 1, 25% are made on Line 2, and 45% are made on Line 3. 3% of items from Line 1 are defective, 5% of items from Line 2 are defective, and 4% of items from line 3 are defective. You select an item at random and observe that it’s defective. What is the probability it was made on Line 2? Example 1 in 10,000 people have a rare disease. The test for this disease gives 5% false positives and 3% false negatives. You take the test, and your result it positive. What is the probability you have the disease?

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