Suppose you are a travel agent. A broker approaches you and offers to pay you a

Question

Suppose you are a travel agent. A broker approaches you and offers to pay you a certain amount of money for organizing events for bus trips the broker advertises. The broker guarantees that you can book 10 buses every month. There are two types of bus trips: Economy and Premium. The broker cannot predict with certainty how many of each bus types there will be, but tells you to expect that on average 1 in 5 buses will be premium buses. Please answer the following questions:

1) Show the probability model for this distribution. 2P

2) What is the expected value for the number of premium buses per month? 1P

3) What is the variance of the number of premium buses per month? 1P

4) What is the probability of the broker sending you 6 or more premium buses per month? 1P

Explanation / Answer

1.

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 =    10
p = the probability of a success = 1/5 = 0.2
x = the number of successes

Hence,

P(10, x) = 10Cx 0.2^x (1 - 0.2)^(10 - x)  

or

P(10, x) = 10Cx 0.2^x 0.8^(10 - x)   [ANSWER]

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

2.

E(x) = n p = 10*0.2 = 2 [ANSWER]

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

3.

Var(x) = n p (1-p) = 10*0.2*(1-0.2) = 1.6 [ANSWER]

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

4.

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 =    10      
p = the probability of a success =    0.2      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.993630618
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.006369382 [ANSWER]

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects