Question

A nationwide award for high school students is given to outstanding students who are sophomores, juniors, or seniors (freshmen are not eligible). Of the award-winners, 65 percent are SENIORS, 20 percent JUNIORS, and 15 percent are SOPHOMORES.

Note: Your answers should be expressed as decimals rounded to three decimal places.

(a) Suppose we select award-winners one at a time and continue selecting until a SENIOR is selected. What is the probability that we will select exactly three award-winners?

(b) Suppose we select award-winners one at a time and continue selecting until a JUNIOR is selected. What is the probability that we will select at least three award-winners?

(c) Suppose we select award-winners one at a time continue selecting until a SOPHOMORE is selected. What is the probability that we will select 2 or fewer award-winners?

Answer 1

• Let X be a random variable denoting the number of trials required to select the first senior.
  The probability of selecting a senior is 0.65.
  Thus, the probability of not seleccting a senior = 1 - 0.65 = 0.35.
  
  In the trial the first senior is selected.
  That is, in the remaining (x-1) trials, seniors are not selected.
  Hence,
  
  
  Here, we are to find:
  
  
  
  
  
• Let Y be a random variable denoting the number of trials required to select the first junior.
  The probability of selecting a senior is 0.2.
  Thus, the probability of not seleccting a senior = 1 - 0.2 = 0.8.
  
  In the trial the first senior is selected.
  That is, in the remaining (y-1) trials, seniors are not selected.
  Hence,
  
  
  Here, we are to find:
  
  
  
  
  
  
  
• Let Z be a random variable denoting the number of trials required to select the first sophomore.
  The probability of selecting a sophomore is 0.15.
  Thus, the probability of not seleccting a senior = 1 - 0.15 = 0.85.
  
  In the trial the first senior is selected.
  That is, in the remaining (z-1) trials, seniors are not selected.
  Hence,
  
  
  Here, we are to find:
  
  
  
  
  

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