PROBABILITY KARNAUGH MAPS WHAT IS A KARNAUGH MAP
PROBABILITY – KARNAUGH MAPS
WHAT IS A KARNAUGH MAP? •
IN A VENN DIAGRAM: • A is everything in the ‘A’ circle • B is everything in the ‘B’ circle • A ∩ B’ means ‘A but not B’ • A ∩ B means ‘A and B’ or ‘overlap/intersection of A and B’ • A’ ∩ B means ‘B but not A’ • A’ ∩ B’ means ‘not A or B’ or ‘everything outside the A and B circles’
IN A KARNAUGH MAP: • We can set these probabilities out in table form • The rows and columns can be added up, for example, if we look at the first column: Pr(A ∩ B) + Pr(A’ ∩ B) = Pr(B) B Not B (B compleme nt) A Not A (A compleme nt) Pr(B) + Pr(B’) = 1 Pr(A) + Pr(A’) = 1
HOW CAN WE USE KARNAUGH MAPS? • They are useful if we have some values out of the table, because we can use them to fill in the gaps B B’ A 0. 2 0. 3 0. 5 A’ 0. 2 0. 3 0. 5 0. 4 0. 6 1 This box should always add to ONE
HOW CAN WE USE KARNAUGH MAPS? • From this, we can answer questions: B B’ A 0. 2 0. 3 0. 5 A’ 0. 2 0. 3 0. 5 0. 4 0. 6 1 • What is Pr(A ∩ B’)? • Pr(A ∩B’) = 0. 3 • Refer back to the template to find the different probabilities
WORDED PROBLEM USING KARNAUGH MAPS • You go to a restaurant where they sell different types of burgers. The probability of choosing a burger at random and getting one with cheese is 0. 67, getting a burger with chicken is 0. 24, and not getting cheese or chicken is 0. 23 • Find the probability that the randomly chosen burger: • a) Has cheese and chicken • b) Has cheese or chicken • c) Has no cheese • d) Has chicken but no cheese
FILLING IN THE KARNAUGH MAP TABLE • You go to a restaurant where they sell different types of burgers. The probability of choosing a burger at random and getting one with cheese is 0. 67, getting a burger with chicken is 0. 24, and not getting cheese or chicken is 0. 23 Cheese Not cheese Chicken 0. 14 0. 1 0. 24 Not chicken 0. 53 0. 23 0. 76 0. 67 0. 33 1
Find the probability that the randomly chosen burger: • a) Has cheese AND chicken Pr(cheese ∩ chicken) = 0. 14 • Remember the addition law of probability: Pr(A U B) = Pr(A) + Pr(B) – Pr(A ∩ B) You will need this formula to help figure out unknown values in some Karnaugh maps b) Has cheese OR chicken Pr (cheese U chicken) = Pr(cheese) + Pr(chicken) – Pr(cheese ∩ chicken) = 0. 67 + 0. 24 – 0. 14 = 0. 77 • c) Has no cheese Pr(cheese’) = 0. 33 • d) Has chicken but no cheese Pr(chicken ∩ cheese’) = 0. 1
QUESTIONS TO DO • Complete the Karnaugh maps worksheet
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