Consider 5 independent Bernoulli's trials each with probability of success p. If the probability of at least one failure is greater than or equal to \[{{31} \over 32},\] then p lies in the interval :
Solutions · 1
Official solution
Here is 5 trials.
So according to Bernoulli trial n = 5
P( at least one failure) = 1 - P( no failure)
According to question,
1 - P( no failure) \[ \ge {{31} \over {32}}\]
\[ \Rightarrow \] 1 - P( x = 5 ) \[ \ge {{31} \over {32}}\]
[Note: no failure = all success]
\[ \Rightarrow \] 1 - \[{}^5{C_5}\,{p^5}\] \[ \ge {{31} \over {32}}\]
\[ \Rightarrow \] 1 - p5 \[ \ge {{31} \over {32}}\]
\[ \Rightarrow \] p5 \[ \le \] \[{1 \over {32}}\]
\[ \Rightarrow \] p \[ \le \] \[{1 \over {2}}\]
\[\therefore\] p \[ \in \] \[\left[ {0,{1 \over 2}} \right]\]
Note: p can not be less than 0 as probability is always \[\ge\] 0 and \[\le\] 1.