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Introduction to Probability

Introduction to Probability. By Dr. Carol A. Marinas. Sample Space. Definition: a SET of all possible outcomes. Example #1: Flipping a coin Sample Space or S = {heads, tails} Example #2: Rolling a die Sample Space or S = { 1, 2, 3, 4, 5, 6}. Probability.

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Introduction to Probability

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  1. Introduction to Probability By Dr. Carol A. Marinas

  2. Sample Space • Definition: a SET of all possible outcomes. • Example #1: Flipping a coin • Sample Space or S = {heads, tails} • Example #2: Rolling a die • Sample Space or S = { 1, 2, 3, 4, 5, 6}

  3. Probability • The probability of an event A is: P(A) = Number of elements of A = n(A) Number of elements of S n(S) Example: Rolling of a die P(getting a 6) = n(sixes) = 1 n(Sample Space) 6

  4. “OR” • When events A and B are mutually exclusive, P(A or B) = P(A U B) = P(A) + P(B) [Mutually exclusive means A ∩ B = Ø.] • Example: P(rolling a 4 or a 6) • Since rolling a 4 or rolling a 6 are mutually exclusive events, P(4 or 6) = P(4) + P(6) = 1/6 + 1/6 = 1/3

  5. “OR” • When events A and B are not mutually exclusive, then P(A U B) = P(A) + P(B) – P(A ∩ B) • P( roll is even or roll is multiple of 3) • Roll is even: 2, 4, 6 • Roll is multiple of 3: 3, 6 • P(roll is even) + P(roll is mult 3) – P (even & mult of 3)= • = 3/6 + 2/ 6 – 1/6 • = 4/6 or 2/3

  6. Other Probabilities _ _ • P(A) + P(A) = 1 so P(A) = 1 – P(A) • P(roll is 5) + P(roll is NOT a 5) = 1 • P(Ø) = 0 (impossible event) • P(roll is greater than 12) = 0 • P(S) = 1 (certain event) • P(roll is less than 7) = 1 • 0 < P(A) < 1

  7. Rolling 2 die • See the Sample Space on Page 382 • The die are white and orange so there are 36 outcomes in the sample space. • Example: P(rolling a sum of 7) • Sums of 7:(1,6), (6,1), (2,5), (5, 2), (3, 4), (4, 3) • P(rolling a sum of 7) = 6/36 or 1/6 • Where are the sums of 7 found in the chart?

  8. More Dice Problems • P(sum is 8) = • P(sum is 2 or sum is 11) = • P(sum is 4 or sum is greater than 10) = • P(sum is multiple of 3 or sum is greater than 8) =

  9. Other Dice Problems • P(sum is 8) = 5/36 • P(sum is 2 or sum is 11) = Sum is 2: (1, 1) Sum is 11: (5, 6), (6,5)Mutually exclusive so P(sum is 2 or sum is 11) = P(sum is 2) + P(sum is 11) = 1/36 + 2/36 = 3/36 = 1/12

  10. Other Dice Problems • P(sum is 4 or sum is greater than 10) = Sum is 4: (1,3), (3, 1), (2, 2) Sum > 10: (5, 6), (6, 5), ( 6, 6) Mutually exclusive so • P(sum is 4 or sum is greater than 10) = P(sum is 4) + P(sum is greater than 10) = 3/36 + 3/36 = 6/36 = 1/6

  11. More Dice Problems • P(sum is multiple of 3 or sum > 8) = Sum is multiple of 3: (1, 2), (2, 1), (5, 1), (1, 5), (2, 4), (4, 2), (3, 3), (6, 3), (3, 6), (5, 4), (4, 5),(6, 6) Sum is greater than 8: (6, 3), (3, 6), (5, 4), (4, 5), (6, 4), (4, 6), (5, 5), ( 6, 5), (5, 6), (6,6) • P(sum is multiple of 3 or sum > 8) = P(sum is mult 3) + P(sum > 8) – P(sum is mult 3 & > 8) = 12/36 + 10/36 – 5/36 = 17/36

  12. Hope you enjoyed Probability

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