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Section 7.2: Probability Models

Section 7.2: Probability Models. Probability Models. A Probability Model describes all the possible outcomes and says how to assign probabilities to any collection of outcomes (events). Example: All possible outcomes listed. All probabilities assigned total 1. Rules of Probability.

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Section 7.2: Probability Models

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  1. Section 7.2: Probability Models

  2. Probability Models • A Probability Model describes all the possible outcomes and says how to assign probabilities to any collection of outcomes (events). • Example: • All possible outcomes listed. • All probabilities assigned total 1.

  3. Rules of Probability • Any probability is a number between 0 and 1. • An event with probability 0 never occurs. • An event with probability 1 occurs on every trial. • All probabilities must add up to a total of 1. • The probability that an event does not happen is 1 minus the probability that the event does happen. • P(Eʹ) = 1 – P(E) • Eʹ is called E prime and is the “complement” of E. E is the event you are checking the probability of. • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities (Addition Rule).

  4. Conditional Probability • Conditional probability is the probability of an event occurring, given that another event has already occurred. • The conditional probability of Event B occurring, given that Event A has occurred is denoted by P(B|A) and is read as “Probability of B, given A”. • Example: Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced)…

  5. Answer… • Because the first card is a king and has not been replaced, there are now only 51 cards. • So the probability that the second card chosen is a queen will be 4/51 or 0.078.

  6. Conditional Probability • Researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given the child has the gene. • P(A|B) = 33/72 = 0.458

  7. Independent or Dependent? • Two events are independent if the occurrence of one does not affect the occurrence of the other (mutually exclusive)… P(B|A) = P(B) or if P(A|B) = P(A) • Events which are not independent are dependent. • Ex., Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B). • P(B|A) = 4/51 • P(A|B) = 4/52 • The occurrence of A changes the probability of the occurrence of B, so the events are dependent. • If the card was replaced, is it still dependent?

  8. Multiplication Rule • The probability that two events A and B will occur in sequence is… • P(A and B) = P(A) * P(B|A) • If events A and B are independent, then the rule can be simplified to… • P(A and B) = P(A) * P(B) • This can be used for any number of events.

  9. Multiplication Rule • Two cards are selected from a standard deck without replacement. Find the probability that both are hearts. • P(H1 and H2) = P(H1) * P(H2|H1) • =13/52 * 12/51 = 156/2652 = 0.059 • A coin is tossed and a die is rolled. Find the probability of getting a tail and rolling a 6. • P(T and 6) = P(T) * P(6) • =1/2 * 1/6 = 1/12 = 0.083

  10. Example • In a box of 11 parts, four of the parts are defective. Two parts are selected without replacement. • Find the probability that both parts are defective. • 4/11 * 3/10 = 12/110 = 0.109 • Find the probability that both parts are not defective. • 7/11 * 6/10 = 42/110 = 0.382 • Find the probability that at least one is defective. • This is the complement of E…1 – 4/11 = 7/11 = 0.636

  11. Work… • Group Work: • White Book: Page 135-137, #11, 12, 17-20 • Homework: • Green Book: Pg. 429-430, #19-23 • Green Book: Pg. 439, #33 • Green Book: Pg. 443, #43

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