1 A company reports that their computer is down an average o

1. A company reports that their computer is \"down\" an average of 1.2 times during an 8-hour shift.

What\'s the probability that
a. the computer will be down 3 times during an 8-hour shift?

b. the computer will not be down during an 8-hour shift (x=0)?

2. Getrich Bank records show that an average of 20 people arrive at a teller\'s counter during an hour.

What\'s the probability that

a. exactly 30 will arrive during a 1-hour period?

b. ten or fewer will arrive during a 1-hour period?

c. more than 10 will arrive during a 1-hour period?

3. The Colorall Paint Factory uses agent A in the paint manufacturing process. There\'s an average of 3 particles of agent A in a cubic foot of the air emitted during the production process. What\'s the probability that

a. there will be five particles of agent A in a cubic foot of air emitted from the factory?

b. there will be no agent A in a cubic foot of air emission?

Solution

I)
Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a)
P( X = 3 ) = e ^-1.2 * 1.2^3 / 3! = 0.0867
b)
P( X = 0 ) = e ^-1.2 * 1.2^0 / 0! = 0.3012

II)
a)
P( X = 30 ) = e ^-20 * 20^30 / 30! = 0.0083

b)
P( X < = 10) = P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-20 * 20 ^ 10 / 10! + e^-20 * 30 ^ 9 / 9! + e^-20 * ^ 8 / 8! + e^-20 * ^ 7 / 7! + e^-20 * ^ 6 / 6! + e^-20 * ^ 5 / 5! + e^-20 * ^ 4 / 4! + e^-20 * ^ 3 / 3! + e^-20 * ^ 2 / 2!
e^-20 * ^ 1 / 1! + e^-20 * ^ 0 / 0!
= 0.0108

c)
P( X > 10) = 1 -P ( X <= 10) = 1 - 0.0108 = 0.9892

III)
a)
P( X = 5 ) = e ^-3 * 3^5 / 5! = 0.1008

b)
P( X = 0 ) = e ^-3 * 3^0 / 0! = 0.0498