An online site reports the probability a 25 year old adult w

An online site reports the probability a 25 year old adult will survive to age 35 is 0.989. You select twenty 25 year old adults at random. What is the probability exactly 19 survive to age of 35? All 20 survive? At least 16 survive?

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.989      
x = the number of successes =    19      
          
Thus, the probability is          
          
P (    19   ) =    0.17830036 [ANSWER]
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B)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.989      
x = the number of successes =    20      
          
Thus, the probability is          
          
P (    20   ) =    0.801541166

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C)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.989      
x = our critical value of successes =    16      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   15   ) =    2.17535E-06
          
Thus, the probability of at least   16   successes is  
          
P(at least   16   ) =    0.999997825 [ANSWER]