Geometric distribution with given probability value.The probability of a man hitting a target is 2/3. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is the least number of shots needed in order to have a probability (of hitting the target) greater than 0.95?

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Geometric distribution with given probability value.The probability of a man hitting a target is 2/3. If he doesn&#039;t stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is the least number of shots needed in order to have a probability (of hitting the target) greater than 0.95?

Emaidedip6g · Accepted Answer

Step 1We want the probability of not hitting to be   &amp;lt;  0.05. The probability of not hitting in x trials is             (              1        3            )        x  . We want this to be less than 0.05.Step 2One does not really need theory to find the answer, just a bit of fooling around with the first few powers of 3. But if we want to use logarithms, we have             (              1        3            )        x    &amp;lt;  0.05 if and only if   x  ln  ⁡  (  1      /    3  )  &amp;lt;  ln  ⁡  (  0.05  ). Now we can find by calculator the solution of   x  ln  ⁡  (  1      /    3  )  =  ln  ⁡  (  0.05  ) and round suitably. (Logarithms to any base will do. You may prefer logarithms to the base 10.)

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