1.
a.
i. Since subsystems A and B are independent, P(A and B) = P(A)*P(B) =
(0.25)*(0.33) = 0.0825 = 8.25%
ii. Again, since subsystems A and B are independent, P(A or B) = P(A)+P(B)
= (0.25)+(0.33) = 0.58 = 58%
iii. If P(A) represents the probability that subsystem A survives at least 10
years, then P(AC) represents the probability that subsystem A fails after 10
years. Since the same could be said for subsystem B, P(AC and BC) =
P(AC)*P(BC) = (1 – 0.25)*(1 – 0.33) = (0.75)(0.67) = 0.5025 = 50.25%
iv. P(only B) = P(B and AC) = P(B)*P(AC) = (0.33)*(0.75) = 0.2475 =
24.75%
b. Bayesian Inference will be used to solve this problem:
Hypotheses:
Subsystem A will survive at least 10 years
Subsystem B will survive at least 10 years
Prior Probabilities:
P(HA) = Probability that subsystem A survives at least 10 years = 0.25
P(HB) = Probability that subsystem B survives at least 10 years = 0.33
After 2 years of observation, we are given the following conditional probabilities:
P ( E|H A )=0.8 P ( E|H B )=0.9
, P ( E|H CA )=0.2 P ( E|H CB ) =0.1
From Bayes Theorem:
P ( H ) P( E∨H )
P ( H|E )=
P ( H ) P ( E|H ) + P ( H ) P(E∨H )
C C
Therefore,
( 0.25)( 0.8) 0.2
P ( H A|E )= = ≈ 0.5714
(0.25)(0.8)+(1−0.25)(0.2) 0.35
and
(0.33)(0.9) 0.297
P ( H B|E )= = ≈ 0.8159
(0.33)( 0.9)+(1−0.33)(0.1) 0.364
c. Due to the large gap between subsystem A and B probabilities just after one revision, it
would be best if the manufacturer invested more in developing subsystem B rather
than splitting resources between both trying to close the statistical gap.
2. Some decision-making biases I would need to be aware of in this project manager
scenario can result from several heuristics, including the availability, representativeness,
anchoring, and adjustment heuristic. Specifically, when assessing probabilities, some biases
resulting from the availability heuristic include inaccurate estimations based on what available
pertinent information is easiest to remember or imagine, skewing our perception away from what
the true probabilities are. Inaccurate estimations can also derive from biases resulting from the
representativeness heuristic, such as when base rate frequencies are not considered or when
unrealistic expectations of sequences appearing random and chance inherently balancing itself
out over time are upheld. Additionally, resulting from the anchoring and adjustment heuristics,
a.
i. Since subsystems A and B are independent, P(A and B) = P(A)*P(B) =
(0.25)*(0.33) = 0.0825 = 8.25%
ii. Again, since subsystems A and B are independent, P(A or B) = P(A)+P(B)
= (0.25)+(0.33) = 0.58 = 58%
iii. If P(A) represents the probability that subsystem A survives at least 10
years, then P(AC) represents the probability that subsystem A fails after 10
years. Since the same could be said for subsystem B, P(AC and BC) =
P(AC)*P(BC) = (1 – 0.25)*(1 – 0.33) = (0.75)(0.67) = 0.5025 = 50.25%
iv. P(only B) = P(B and AC) = P(B)*P(AC) = (0.33)*(0.75) = 0.2475 =
24.75%
b. Bayesian Inference will be used to solve this problem:
Hypotheses:
Subsystem A will survive at least 10 years
Subsystem B will survive at least 10 years
Prior Probabilities:
P(HA) = Probability that subsystem A survives at least 10 years = 0.25
P(HB) = Probability that subsystem B survives at least 10 years = 0.33
After 2 years of observation, we are given the following conditional probabilities:
P ( E|H A )=0.8 P ( E|H B )=0.9
, P ( E|H CA )=0.2 P ( E|H CB ) =0.1
From Bayes Theorem:
P ( H ) P( E∨H )
P ( H|E )=
P ( H ) P ( E|H ) + P ( H ) P(E∨H )
C C
Therefore,
( 0.25)( 0.8) 0.2
P ( H A|E )= = ≈ 0.5714
(0.25)(0.8)+(1−0.25)(0.2) 0.35
and
(0.33)(0.9) 0.297
P ( H B|E )= = ≈ 0.8159
(0.33)( 0.9)+(1−0.33)(0.1) 0.364
c. Due to the large gap between subsystem A and B probabilities just after one revision, it
would be best if the manufacturer invested more in developing subsystem B rather
than splitting resources between both trying to close the statistical gap.
2. Some decision-making biases I would need to be aware of in this project manager
scenario can result from several heuristics, including the availability, representativeness,
anchoring, and adjustment heuristic. Specifically, when assessing probabilities, some biases
resulting from the availability heuristic include inaccurate estimations based on what available
pertinent information is easiest to remember or imagine, skewing our perception away from what
the true probabilities are. Inaccurate estimations can also derive from biases resulting from the
representativeness heuristic, such as when base rate frequencies are not considered or when
unrealistic expectations of sequences appearing random and chance inherently balancing itself
out over time are upheld. Additionally, resulting from the anchoring and adjustment heuristics,
