7. A discrete random variable X has a binomial distribution with n =15 and p =0.

Question

7. A discrete random variable X has a binomial distribution with n=15 and p=0.2.

(1)Find P(X <5). (2)Find P(X >5). (3)Find P(5< X <8). (4)Find P(5 X 8).

8. A prescription drug manufacturer claims that only 10% of all new drugs that are shown to be effective in animal tests ever pass through all the additional testing required to be marketed. The manufacturer currently has 20 new drugs that have been shown to be effective in animal tests and they await for further testing and approval. Let X be the number of marketed drugs among 20.

(1) What is the probability function (probability distribution) of the random variable X?

(2) Find the probability that no more than 3 drugs are marketed.

(3) Find the probability that more than 3 drugs are marketed.

(4) Find the probability that exactly 3 drugs are marketed.

(5) Find the expected value and the standard deviation of X.

x

2

3

4

5

6

p(x)

0.05

0.15

0.20

0.35

c

x

2

3

4

5

6

p(x)

0.05

0.15

0.20

0.35

c

Explanation / Answer

7.

a)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.2      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.835766276
          
Which is also          
          
P(fewer than   5   ) =    0.835766276 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.2      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   5   ) =    0.93894857
          
Thus, the probability of at least   6   successes is  
          
P(more than   5   ) =    0.06105143 [ANSWER]

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c)

Hence, it is between 6 and 7 inclusive, as 5 and 8 are not included.

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    6      
x2 =    7      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.2      
          
Then          
          
P(at most    5   ) =    0.93894857
P(at most    7   ) =    0.99576025
          
Thus,          
          
P(between x1 and x2) =    0.05681168   [ANSWER]

*************************

d)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    5      
x2 =    8      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.2      
          
Then          
          
P(at most    4   ) =    0.835766276
P(at most    8   ) =    0.999215015
          
Thus,          
          
P(between x1 and x2) =    0.163448739   [ANSWER]  
  

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