October 3, 2011
Poisson Distribution - Solutions
Example 1
The quality control manager of Marilyn’s Cookie is inspecting a batch of chocolate chip cookies that have
just been baked. If the production process is in control, the mean number of chip parts per cookie is 6.0
What is the probability that that in any particular cookie being inspected,
A.
B.
C.
D.
Less than 5 chips will be found
Exactly 5 chips will be found
5 or more chips will be found
Either 4 or 5 chips will be found.
Solution
λ=6
X
0
1
2
3
4
5
…
P(X)
0.00248
0.01487
0.04462
0.08924
0.13385
0.16062
…
A. 𝑃(𝑋 < 5) = 𝑃(𝑋 = 0) + 𝑃(𝑋 = 1) + 𝑃(𝑋 = 3) + 𝑃(𝑋 = 4) = 0.285057
B. 𝑃(𝑋 = 5) = 0.16062
C. 𝑃(𝑋 ≥ 5) = 1 − [𝑃(𝑋 = 0) + 𝑃(𝑋 = 1) + ⋯ + 𝑃(𝑋 = 4)] = 1 − 0.285057 = 0.7149
D. 𝑃(𝑋 = 4 & 𝑋 = 5) = 𝑃(𝑋 = 4) + 𝑃(𝑋 = 5) = 0.13385 + 0.16062 = 0.29447
Example 2
The U.S. transportation maintains a statistic for mishandled bags per 1000 airline passengers. In 2007,
the airline mishandled 7 bags per 1000 passengers. What is the probability that in the next 1000
passengers, airlines will have
A. No mishandled bags?
B. At least one mishandled bag?
C. At least two mishandled bags?
Solution
𝜆=7
X
0
1
2
…
P(X)
0.0009
0.0064
0.0223
…
A. 𝑃(𝑋 = 0) = 0.0009
B. 𝑃(𝑋 ≥ 1) = 1 − 𝑃(𝑋 = 0) = 1 − 0.0009 = 0.9991
C. 𝑃(𝑋 ≥ 2) = 1 − [𝑃(𝑋 = 0) + 𝑃(𝑋 = 1)] = 1 − [0.0009 + 0.0064] = 0.9927
Example 3
Based on past experience, it is assumed that the number of flaws per foot in rolls of grade 2 paper
follows a Poisson distribution with a mean of 1 flaw per 5 feet of paper (0.2 flaws per foot). What is the
probability that in a
A. 1 - foot roll, there will be at least 2 flaws?
B. 12 – foot roll there will be at least 1 flaw?
C. 50 – foot roll there will be at least 5 flaws, but no more than 10 flaws?
Solution
A. 𝜆 = 0.2 𝑃(𝑋 ≥ 2) = 1 − [𝑃(𝑋 = 0) + 𝑃(𝑋 = 1)]
= 1 − [𝑒 −0.2 + 𝑒 −0.2 (0.2)]
= 1 − [0.8187 + 0.1637]
= 0.0176
B. 𝜆 = 0.2(12) = 2.4
𝑃(𝑋 ≥ 1) = 1 − [𝑃(𝑋 = 0)]
= 1 − 𝑒 −2.4
= 0.9093
C. 𝜆 = 0.2(50) = 10
𝑃(5 ≤ 𝑋 ≤ 10) = 𝑃(𝑋 = 5) + 𝑃(𝑋 = 6) + 𝑃(𝑋 = 7) + ⋯ + 𝑃(𝑋 = 10)
𝑒 −10 105 𝑒 −10 106 𝑒 −10 107 𝑒 −10 108 𝑒 −10 109 𝑒 −10 1010
=
+
+
+
+
+
5!
6!
7!
8!
9!
10!
= 0.0378 + 0.0630 + 0.0901 + 0.1126 + 0.1251 + 0.1251
= 0.5537