Events with two possible outcomes are modelled by the following equation: if the probability of success is [tex]p[/tex], you have [tex]n[/tex] trials, and X is the random variable representing the number of times the event with probability p happens, you have
[tex]\displaystyle P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}[/tex]
Now, you want X=0 (i.e. the event never happens). So, you have
[tex]\displaystyle P(X=0) = \binom{n}{0} p^0 (1-p)^{n-0} = (1-p)^n[/tex]
[tex](1-p)^n[/tex] is the probability that the event does not occur in any of the trials.
Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
Events with two possible outcomes are modelled by the following equation: if the probability of success is, you have trials, and X is the random variable representing the number of times the event with probability p happens, you have
[tex]P(x=k) =\left[\begin{array}{ccc}n\\k\\\end{array}\right] p^k ( 1-p)^{n-k}[/tex]
Now, you want X=0 (i.e. the event never happens). So, you have
[tex]P(x=0) =\left[\begin{array}{ccc}n\\0\\\end{array}\right] p^0 ( 1-p)^{n-0}=( 1-p)^n[/tex]
Therefore [tex](1-p)^n[/tex] is the probability that the event does not occur in any of the trials.
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