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Mathematics 146 second homework
due Tuesday, May 13, 2014
please show any relevant work to get credit for each problem
3. Suppose that a test for a certain disease
• returns a positive result for 95% of people with the disease, and a negative result for
5% of the people with the disease
• returns a negative result for 90% of the people without the disease, and a positive result
for 10% of the people without the disease
Assume that 30% of the large population actually has the disease.
Use the abbreviations
T = event that a person tested positive for the disease
D = event that a person actually has the disease
Use the information to compute
(a) P (T | D)
(b) P (T 0 | D0 )
(c) P (D | T )
(d) P (D0 | T )
(e) P (D0 | T 0 )
(f) are T and D independent events ?
(g) the probability that a group of five randomly chosen people from this population are
all disease-free.
(h) the probability that in a group of ten randomly chosen people from this population
exactly three are disease-free.
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4. A particular surgery is successful for 70% of the patients. In a group of 15 patients
undergoing this surgery
(a) calculate (or determine) the probability that exactly 10 had successful surgeries
(b) if x represents the number of successful surgeries in this group of 15 patients, sketch
the probability distribution of x
(c) what is the mean value µ of x ?
(d) what is the standard deviation σ of x ?
(e) what percentage of the x-values are within one standard deviation of the mean ?
(between µ − σ and µ + σ)
• what percentage of the x-values are within two standard deviation of the mean ?
(between µ − 2σ and µ + 2σ)
5. Suppose the probability of any day being pleasant is 90%. In a 50-day period calculate
(a) the probability of exactly 45 days being pleasant
(b) the probability of at least 45 days being pleasant
(c) the probability of the 46th day being pleasant assuming that the preceding 45 days had
been pleasant.