Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion MATH 105: Finite Mathematics 8-2: The Binomial Probablity Model Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Outline Introduction to Bernoulli Processes 1 Bernoulli Trials and the Bernoulli Probability Formula 2 Examples 3 Conclusion 4
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Outline Introduction to Bernoulli Processes 1 Bernoulli Trials and the Bernoulli Probability Formula 2 Examples 3 Conclusion 4
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion A Motivating Example Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[ H ] = 2 5 is flipped two times. Find the probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three times. Find the probability of exactly one Heads. Example The same unfiar coin as in the previous examples is flipped four times. Find the probability of exactly one Heads.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion A Motivating Example Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[ H ] = 2 5 is flipped two times. Find the probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three times. Find the probability of exactly one Heads. Example The same unfiar coin as in the previous examples is flipped four times. Find the probability of exactly one Heads.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion A Motivating Example Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[ H ] = 2 5 is flipped two times. Find the probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three times. Find the probability of exactly one Heads. Example The same unfiar coin as in the previous examples is flipped four times. Find the probability of exactly one Heads.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion A Motivating Example Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[ H ] = 2 5 is flipped two times. Find the probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three times. Find the probability of exactly one Heads. Example The same unfiar coin as in the previous examples is flipped four times. Find the probability of exactly one Heads.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Making the Process More Complicated Example Now suppose that you flip the coin four times and wish to find the probability of getting exactly two heads. What about getting exactly three heads? Exactly four heads? Note: A pattern emerges when we repeat the same action, flipping the coin, multiple times. The Bernoulli Probability Formula gives us a way to quickly compute such probabilities.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Making the Process More Complicated Example Now suppose that you flip the coin four times and wish to find the probability of getting exactly two heads. What about getting exactly three heads? Exactly four heads? Note: A pattern emerges when we repeat the same action, flipping the coin, multiple times. The Bernoulli Probability Formula gives us a way to quickly compute such probabilities.
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Outline Introduction to Bernoulli Processes 1 Bernoulli Trials and the Bernoulli Probability Formula 2 Examples 3 Conclusion 4
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Bernoulli Trials In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C ( n , r )( p ) r (1 − p ) n − r
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Outline Introduction to Bernoulli Processes 1 Bernoulli Trials and the Bernoulli Probability Formula 2 Examples 3 Conclusion 4
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Example Find the probability of 3 successes in 4 trials with Pr[ success ] = 1 3 � 1 � 3 � 2 � 1 � 1 � � 2 � C (4 , 2) = 4 3 3 27 3 � 2 � = 4 81 = 8 81
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Example Find the probability of 3 successes in 4 trials with Pr[ success ] = 1 3 � 1 � 3 � 2 � 1 � 1 � � 2 � C (4 , 2) = 4 3 3 27 3 � 2 � = 4 81 = 8 81
Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion Taking a Quiz Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the: 1 probability he scores 7 out of 10. 2 probablility he scores 8 or better. 3 probability he fails (6 or less).
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