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17 March, 15:38

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y. (Enter your answers in set notation.)

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  1. 17 March, 17:13
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    The outcomes corresponding to the five smallest possible values of Y are given by this set

    { (S, S, S), (F, S, S, S), (F, F, S, S, S), (S, F, S, S, S), (F, F, F, S, S, S), (F, S, F, S, S, S), (S, F, F, S, S, S), (S, S, F, S, S, S), (F, F, F, F, S, S, S), (F, F, S, F, S, S, S), (F, S, F, F, S, S, S), (F, S, S, F, S, S, S), (S, F, F, F, S, S, S), (S, F, S, F, S, S, S), (S, S, F, F, S, S, S) }

    Step-by-step explanation:

    You need to end with 3 success (the experiment ends there). The smallest possible value Y can take is 3, and it is given by

    (S, S, S)

    For Y = 4 we have only 1 possibility, the first trial is a failure and the following 3 ones are success

    (F, S, S, S)

    For Y = 5 we have

    (F, F, S, S, S) and

    (S, F, S, S, S)

    For Y = 6

    (F, F, F, S, S, S)

    (F, S, F, S, S, S)

    (S, F, F, S, S, S)

    (S, S, F, S, S, S)

    And lastly, for Y = 7, we have

    (F, F, F, F, S, S, S)

    (F, F, S, F, S, S, S)

    (F, S, F, F, S, S, S)

    (F, S, S, F, S, S, S)

    (S, F, F, F, S, S, S)

    (S, F, S, F, S, S, S)

    (S, S, F, F, S, S, S)

    Therefore, the total amount of possible outcomes corresponding to the five smallest possible values is

    { (S, S, S), (F, S, S, S), (F, F, S, S, S), (S, F, S, S, S), (F, F, F, S, S, S), (F, S, F, S, S, S), (S, F, F, S, S, S), (S, S, F, S, S, S), (F, F, F, F, S, S, S), (F, F, S, F, S, S, S), (F, S, F, F, S, S, S), (F, S, S, F, S, S, S), (S, F, F, F, S, S, S), (S, F, S, F, S, S, S), (S, S, F, F, S, S, S) }
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