Introduction to Statistics: Difference between revisions
wikademia>Jimmytharpe →Probability: Added some narrations of equations to make understanding easier. More should still be added IMO. |
wikademia>Jimmytharpe |
||
| Line 62: | Line 62: | ||
where <math>N_{A}</math> is the number of times event <math>A</math> occurred, and <math>N</math> is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event <math>A</math> equals the number of times event <math>A</math> occurs divided by the number of times the experiment was repeated (or the number of times event <math>A</math> ''could have'' occurred)." As <math>N</math> approaches infinity, the fraction above approaches the true probability of the event <math>A</math>. The value of <math>P(A)</math> is clearly between 0 and 1. If our event is the ''certain event'' <math>\Omega</math>, then for each time we perform the experiment, the event <math>\Omega</math> is observed; <math>N_{\Omega} = N</math> and <math>P(\Omega)=1</math>. If our event is the ''impossible event'' <math>\varnothing</math>, we know <math>N_{\varnothing}=0</math> and <math> P(\varnothing) = 0</math>. | where <math>N_{A}</math> is the number of times event <math>A</math> occurred, and <math>N</math> is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event <math>A</math> equals the number of times event <math>A</math> occurs divided by the number of times the experiment was repeated (or the number of times event <math>A</math> ''could have'' occurred)." As <math>N</math> approaches infinity, the fraction above approaches the true probability of the event <math>A</math>. The value of <math>P(A)</math> is clearly between 0 and 1. If our event is the ''certain event'' <math>\Omega</math>, then for each time we perform the experiment, the event <math>\Omega</math> is observed; <math>N_{\Omega} = N</math> and <math>P(\Omega)=1</math>. If our event is the ''impossible event'' <math>\varnothing</math>, we know <math>N_{\varnothing}=0</math> and <math> P(\varnothing) = 0</math>. | ||
If <math>A</math> and <math>B</math> are ''disjoint events'', then whenever event <math>A</math> is observed, then it is impossible for event <math>B</math> to be observed simultaneously. Therefore the number of times | If <math>A</math> and <math>B</math> are ''disjoint events'', then whenever event <math>A</math> is observed, then it is impossible for event <math>B</math> to be observed simultaneously. Therefore the number of times events <math>A</math> and <math>B</math> occurs are equal to the number of times event <math>A</math> occurred plus the number of times <math>B</math> occurs. This can be expressed as: <br> | ||
<math>N(A\cup B) = N(A) + N(B)</math><br> | <math>N(A\cup B) = N(A) + N(B)</math><br> | ||
Given our definition of probability, we can arrive at the following:<br> | Given our definition of probability, we can arrive at the following:<br> | ||