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Notes #12 Conditional Probability with Venn diagrams

Notes #12 Conditional Probability with Venn diagrams. Conditional Probability. These are the probabilities calculated on the basis that something has already happened. Formula for Conditional Probability. Meaning: P( A |B ) , meaning P( A happening given B occurred ). Problem #1.

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Notes #12 Conditional Probability with Venn diagrams

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  1. Notes #12Conditional Probability with Venn diagrams

  2. Conditional Probability • These are the probabilities calculated on the basis that something has already happened

  3. Formula for Conditional Probability Meaning: P(A|B), meaning P(A happening given B occurred)

  4. Problem #1 • A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test? Set-up: • Create a Key. P(A) = Second Test ; P(B) = First Test 2) Set up Formula P(A|B)= P(A n B)/P(B) P(Second|First) = P(First and Second) P(First) =   0.25/0.42 =   0.60 ANS: 60% passed the first test who also passed the second test.

  5. Problem # 2 0.3 0.4 P(A)=0.3 ; P(B)= 0.4 and P(A|B)=0.5 Find 1. P(A n B) 2. P(A U B) 3. P(A|B’ ) Fill in the Venn diagram A B 0.2 0.2 0.1 0.5 1. P(A|B) is 0.5 so ….. P(A|B) = P(A n B)/P(B) 0.5 = P(A n B) / 0.4 P(A n B) =0.5x 0.4 = 0.2

  6. Problem 2 Continued 0.3 0.4 P(A)=0.3 ; P(B)= 0.4 and P(A|B)=0.5 Find 1. P(A n B) = 0.2 2. P(A U B) 3. P(A|B’ ) Fill in the Venn diagram A B 0.2 0.2 0.1 0.5 2. P(A U B) = P(A) + P(B) – P(A n B) = 0.3 + 0.4 – 0.2 = 0.5

  7. Problem 2 Continued 0.3 0.4 P(A)=0.3 ; P(B)= 0.4 and P(A|B)=0.5 Find 1. P(A n B) = 0.2 2. P(A U B) = 0.5 3. P(A|B’ ) Fill in the Venn diagram A B 0.2 0.2 0.1 0.5 3. P(A|B’) means P(A given Not B) P(A|B’) = P(A n B’)/P(B’) = (0.3-0.2)/(1-0.4) = 0.1/0.6 = 0.1667

  8. Questions Use your diagram to find the probability of choosing a student who, 120 students in a school can opt for one or two languages for IB. 75 choose Japanese and 35 choose Chinese. 20 do neither. b) takes only Japanese, a) Draw a Venn diagram to show this information. c) takes both languages, d) takes only Japanese given that they take at least one language, e) takes Chinese, given that they take Japanese.

  9. P(A|B) = P(Agiven B has occurred) If B has already happened then our event must be somewhere in B BUT, How can A happen if our event must be in the B space ? We can only be in the following Space on our Venn Diagram B B B A A A And so Our Probability P(A|B) is the ratio of Green Space ÷ Red space

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