Suppose we perform 2 experiments and this first experiment can result in any one of m possible outcomes. Suppose the first experiment results in outcome i. Then, the second experiment can result in any of n i possible outcomes i = 1 , . . , m. Q s : what is the number of possible outcomes of the two experiments?Attempt:If we have 1 , . . . , m possible outcomes and say this experiment results in one of them say i. and so second experiment can result in n i for each i so by multiplication principle we have n 1 ⋅ n 2 ⋅ . . ⋅ n m possible outcomes. Is this correct?

Question

Suppose we perform 2 experiments and this first experiment can result in any one of   m possible outcomes. Suppose the first experiment results in outcome   i. Then, the second experiment can result in any of       n    i   possible outcomes   i  =  1  ,  .  .  ,  m.       Q    s    :   what is the number of possible outcomes of the two experiments?Attempt:If we have   1  ,  .  .  .  ,  m possible outcomes and say this experiment results in one of them say   i. and so second experiment can result in       n    i   for each   i so by multiplication principle we have       n    1    ⋅      n    2    ⋅  .  .  ⋅      n    m   possible outcomes. Is this correct?

Jasper Parsons · Accepted Answer

To have the outcomes implied by       n    1   in the second experiment, we need outcome 1 in the first experiment (      E    1    =  1). Similarly we need outcome       E    1    =  2 to get to those       n    2   possibilities, etc.So the possible second-experiment outcomes each depend on a given first-experiment result and the total outcome options are       ∑    1    m    (  1  ⋅      n    i    )  =      ∑    1    m        n    i