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Probability and Statistics Chapter 3 Notes Section 3-2 I. Conditional Probability A 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 π(π΅|π΄) and is read as βprobability of B, given A.β II. Independent and Dependent Events Two events are if the occurrence of one of the events affect the probability of the occurrence of the other event. Two events A and B are if π(π΅|π΄) = π(π΅), or if π(π΄|π΅) = π(π΄). In other words, event B is likely to occur whether event A has or not. Events that are not independent are . To determine if A and B are independent, first calculate , the probability of event . Then calculate , the probability of , given . If the two probabilities are , the events are . If the two probabilities are not , the events are . III. The Multiplication Rule π(π΄ πππ π΅) = π(π΄) β π(π΅|π΄) If Events A and B are dependent; 1) Find the probability the first event occurs 2) Find the probability the second event occurs given the first event has occurred. 3) Multiply these two probabilities to find the probability that both events will occur in sequence. If Events A and B are independent, π(π΄ πππ π΅) = π(π΄) β π(π΅). This simplified rule can be extended for any number of independent events, just like the Fundamental Counting Principle could be extended. Example 1 (Page 149) 1) 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 (itβs not put back into the deck before the queen is drawn. 2) The table shows the results of a study in which researchers examined a childβs IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene. High IQ Normal IQ Total Gene Present 33 39 72 Gene Not Present 19 11 30 Total 52 50 102 SOLUTIONS: 1) The probability of drawing a queen after a king (or any other card) has been taken out of the deck is , or about . 2) There are children who have the gene. They are our . Of these, have a high IQ. So, π(π΅|π΄) = β . Example 2 (Page 150) Decide whether the events are independent or dependent. 1) Selecting king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B). 2) Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B). SOLUTIONS: 1) The probability of pulling a queen out of the deck is . The probability of pulling a queen out of the deck after a king has already been removed is . Since these probabilities are , the events are . 2) The probability of obtaining a 6 is . The probability of obtaining a 6 given that the coin came up heads is . Since these probabilities are , the events are . Example 3 (Page 151) 1) Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen. 2) A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6. SOLUTIONS: 1) Because the first card is not replaced, the events are . π(πΎ πππ π) = * * = β 2) The two events are . π(π» πππ 6) = π(π») β π(6) * = β Example 4 (Page 152) 1) A coin is tossed and a die is rolled. Find the probability of getting a tail and then rolling a 2. 2) The probability that a particular knee surgery is successful is 0.85. Find the probability that three surgeries in a row are all successful. 3) Find the probability that none of the three knee surgeries is successful. 4) Find the probability that at least one of the three knee surgeries is successful. SOLUTIONS 1) The probability of getting a tail and then rolling a 2 is π(π πππ 2) = π(π) β π(2) * = β 2) The probability that each knee surgery is successful is 0.85. Since these events are , the probability that all three are successful is found by their probabilities together. β The probability that all three surgeries are successful is about . 3) Because the probability of success for one surgery is .85, the probability of failure for one surgery is . This is true because failure is the of success. The probability that each knee surgery is not successful is . Since these events are , the probability that all three are not successful is found by their probabilities together. β The probability that none of the surgeries are successful is about . 4) The phrase βat least oneβ means . The complement to the event βat least one is successfulβ is the event β β. Using the complement rule, we can simply the probability that none were successful from to find the probability that at least one was successful. = . The probability that at least one of the surgeries is successful is about . Example 5 (Page 153) More than 15,000 medical school seniors applied to residency programs in 2007. Of those, 93% were matched to a residency position. 74% percent of the seniors matched to a residency position were matched to one of their top two choices. 1) Find the probability that a randomly selected senior was matched to a residency program and it was one of the seniorβs top two choices. 2) Find the probability that a randomly selected senior that was matched to a residency program did not get matched with one of the seniorβs top two choices. 3) Would it be unusual for a randomly selected senior to result in a senior that was matched to a residency position and it was one of the seniorβs top two choices? SOLUTION: 1) These two events are : π(π΄ πππ π΅) = π(π΄) β π(π΅|π΄) = * β . 2) To find this probability, use the complement: π(π΅β² |π΄) = 1 β π(π΅|π΄) = 1 = . 3) : This event occurs around % of the time.