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Probability

Probability. The chance of an event occurring. Symbol: P(x). Probability Experiment. A chance process that leads to well-defined results called outcomes. Example: Roll a die. Sample Space. The set of all possible outcomes of a probability experiment Symbol: S.

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Probability

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  1. Probability The chance of an event occurring. Symbol: P(x)

  2. Probability Experiment A chance process that leads to well-defined results called outcomes. Example: Roll a die

  3. Sample Space The set of all possible outcomes of a probability experiment Symbol: S Example: When rolling a die The sample space is S = {1, 2, 3, 4, 5, 6}

  4. Find the sample space of the Following probability experiments A coin is flipped: Two coins are flipped: {heads, tails} {HH, HT, TH, TT}

  5. Outcome The result of a single trial in a Probability experiment Example: When rolling a die The outcome could be {4}

  6. Tree Diagram A device consisting of line segments originating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment

  7. Tree Diagram Two dice are rolled. Describe the sample space. Start 1st roll 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 2nd roll 36 outcomes

  8. Event Consists of a set of outcomes of a Probability experiment (a subset of The sample space) Example: If you wanted to roll an even Number on a die {2, 4, 6}

  9. Find the event of the Following probability experiments Getting heads twice When flipping 3 coins: Getting a heads when Flipping one coin: Sample Space {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} Sample Space: {heads, tails} Event: {heads} Event: {HHT, HTH, THH}

  10. Simple Event Event with one outcome Example: Flipping 3 coins and getting Tails 3 times {TTT}

  11. Compound Event Event with more than one outcome Example: Flipping a coin 3 times and getting heads twice {HHT, HTH, THH}

  12. Classical Probability Uses sample spaces to determine the numerical probability that an event will happen *Assumes that all outcomes in the sample space are equally likely to occur

  13. Classical Probability The probability of any event E is: ____Number of outcomes in E____ Total # of outcomes in the sample space

  14. Classical Probability Example: For a card drawn from an ordinary deck, Find the probability of getting a queen Since there are 52 cards in a deck and there are 4 queens: P(queen) = 4/52 = 1/13

  15. Classical Probability Example: If you are rolling a die find the probability of the following: P(1) = P(1 or 3) = P(odd) = P(greater than 3) = P(7) = 1/6 2/6 = 1/3 3/6 = 1/2 3/6 = 1/2 0

  16. Probability Rule #1 The probability of any event E is a number (either a fraction or a decimal) between and including 0 and 1. This is denoted by 0 ≤ P(E) ≤ 1

  17. Probability Rule #2 If an event E cannot occur, its probability is 0

  18. Probability Rule #3 If an event E is certain, then the probability of E is 1

  19. Probability Rule #4 The sum of the probabilities of all the outcomes in the sample space is 1.

  20. Two dice are rolled and the sum is noted. 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Find the probability the sum is 4 P(4) = 3/36 = 1/12 Find the probability the sum is 11 P(11) = 2/36 = 1/18 Find the probability the sum is 4 or 11 P(4 or 11) = 5/36

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