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Maths / Probability / Probability of Event not Happening

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Probability of Event not Happening

Probability of Event Not Happening

Suppose a random experiment results in n equally likely outcomes out of which m are favorable to the occurrence of an event A,Then the number of unfavorable outcomes will be:

Number of unfavorable outcomes = Total number of Outcomes - Number of favorable outcomes = n - m

If we denote the probability of occurrence of event A by P(A), then the probability of not occurrence of the event is denoted by P(A')

$P(A^I)=frac{Number; of; unfavourable; outcomes}{Total; number ;of; outcomes}=frac{n-m}{n}$

$P(A^I)=frac{n}{n}- frac{m}{m}=1-frac{m}{n}= 1-P(A)$

$As; 0leq P(A)leq 1;;Rightarrow 0 leq P(A^I)leq 1$

Illustration: An experiment was done 340 times in a day. 160 outcomes from the experiment were favorable, what is the probability that the next outcome is unfavorable.

Solution: Total number of trials = 340

No of favorable trials= 160

No of unfavorable trials= 340-160= 180

$P(E^I)=frac{Number ;of ; unfavourable ;trials}{The; total; number; of; trials}=frac{180}{340}=frac{18}{34}=frac{9}{17}$

We can solve in another way:

$P(E)=frac{Number ;of ;favourable ;trials}{The; total; number; of; trials}=frac{160}{340}=frac{16}{34}=frac{8}{17}$

$P(E^I)=1- P(E)= 1-frac{8}{17}= frac{17-8}{17}=frac{9}{17}$

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