At an outpatient mental health clinic appointment cancellati

At an outpatient mental health clinic, appointment cancellations occur at a mean rate of 2.8 per day on a typical Wednesday. Let X be the number of cancellations on a particular Wednesday. Justify the use of the Poisson model. Cancellations are not independent Cancellations are independent and similar to arrivals Most likely cancellations arrive independently What is the probability that no cancellations will occur on a particular Wednesday? (Round your answer to 4 decimal places.) Probability What is the probability that one cancellations will occur on a particular Wednesday? (Round your answer to 4 decimal places.) Probability What is the probability that more than four cancellations will occur on a particular Wednesday? (Round your answer to 4 decimal places.) Probability What is the probability that four or more cancellations will occur on a particular Wednesday? (Round your answer to 4 decimal places.) Probability

Solution

a)

OPTION B: Cancellations are independent

b)

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    2.8      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.060810063 [answer]

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

c)

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    2.8      
          
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.170268175 [answer]

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

d)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    2.8      
          
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   4   ) =    0.847676057
          
Thus, the probability of at least   5   successes is  
          
P(more than   4   ) =    0.152323943 [ANSWER]

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

E)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    2.8      
          
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.691937433
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.308062567 [answer]