(a) Who do you think is right or wrong and why?
• Sara’s Perspective
Sara expresses concern over the drawing of the number 1000, viewing it as a
suspicious choice because she perceives it as an "easy" or significant number.
1
,However, this line of reasoning is flawed from a statistical standpoint. In a
random raffle, each number from 1 to 1000 has an equal probability of being
selected. Sara’s intuition about
the number 1000 being special does not alter the fact that every ticket has the
same chance of winning.
• Mahra’s Perspective
Mahra presents a valid argument by stating that every ticket, including
number 1000, has an equal probability of being drawn. Thus, the likelihood of
selecting
number 1000 is identical to that of any other number within the range of 1 to
1000.
• Ahmad’s Perspective
Ahmad introduces the notion of potential foul play in the drawing process.
While this reflects a personal belief rather than a statistical argument, it lacks
a
foundation in solid evidence. Although he suggests a very slight chance of
manipulation based on intuition, his reasoning could be more robust if he
employed Bayesian analysis
to consider prior probabilities and evidence related to rigging.
(b) Posterior Probability of Foul Play
We can calculate the posterior probability using Bayes’ Theorem. Bayes’
Theorem. Firstly, we are going to define each values as follows:
The prior probability that the raffle is rigged:
P (FP) = p
If the raffle is rigged, the probability of drawing ticket 1000 is certain
(100percent):
P (T − 1000 | FP) = 1
2
, The probability of drawing ticket 1000 in a fair raffle is 1 in 1000:
1
P (T − 1000 | FR ) =
1000
The total probability of drawing ticket 1000, which can be calculated using the
law of total probability:
P (T − 1000)
By Bayes’ Theorem, we have:
P (T − 1000 | FP) × P (FP)
Prob (FP | T − 1000 ) =
P (T − 1000)
P (T − 1000) = P (T − 1000 | FP) · P (FP) + P (T − 1000 | FR) · P (FR)
1
P (T − 1000) = 1 · p + · (1 − p)
1000
Thus, the posterior probability becomes:
p
P (FP | T − 1000 ) = (1−p)
p+ 1000
(c) Assess Sara’s Strong Disbelief in Light of 1b
Sara’s intense skepticism stems from her emotional reaction to the number
1000, which she perceives as significant or "easy." However, when we analyze
the
situation using the formula from part 1b, it becomes clear that the posterior
probability of foul play is only considerable if the prior probability was already
substantial.
Thus, while Sara’s feelings of suspicion are understandable on an emotional
level, they do not hold up under statistical scrutiny. Her concerns are not
supported by
evidence that would indicate a higher likelihood of rigging in the raffle.
(d) Assess Mahra’s Argument in Light of 1b
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• Sara’s Perspective
Sara expresses concern over the drawing of the number 1000, viewing it as a
suspicious choice because she perceives it as an "easy" or significant number.
1
,However, this line of reasoning is flawed from a statistical standpoint. In a
random raffle, each number from 1 to 1000 has an equal probability of being
selected. Sara’s intuition about
the number 1000 being special does not alter the fact that every ticket has the
same chance of winning.
• Mahra’s Perspective
Mahra presents a valid argument by stating that every ticket, including
number 1000, has an equal probability of being drawn. Thus, the likelihood of
selecting
number 1000 is identical to that of any other number within the range of 1 to
1000.
• Ahmad’s Perspective
Ahmad introduces the notion of potential foul play in the drawing process.
While this reflects a personal belief rather than a statistical argument, it lacks
a
foundation in solid evidence. Although he suggests a very slight chance of
manipulation based on intuition, his reasoning could be more robust if he
employed Bayesian analysis
to consider prior probabilities and evidence related to rigging.
(b) Posterior Probability of Foul Play
We can calculate the posterior probability using Bayes’ Theorem. Bayes’
Theorem. Firstly, we are going to define each values as follows:
The prior probability that the raffle is rigged:
P (FP) = p
If the raffle is rigged, the probability of drawing ticket 1000 is certain
(100percent):
P (T − 1000 | FP) = 1
2
, The probability of drawing ticket 1000 in a fair raffle is 1 in 1000:
1
P (T − 1000 | FR ) =
1000
The total probability of drawing ticket 1000, which can be calculated using the
law of total probability:
P (T − 1000)
By Bayes’ Theorem, we have:
P (T − 1000 | FP) × P (FP)
Prob (FP | T − 1000 ) =
P (T − 1000)
P (T − 1000) = P (T − 1000 | FP) · P (FP) + P (T − 1000 | FR) · P (FR)
1
P (T − 1000) = 1 · p + · (1 − p)
1000
Thus, the posterior probability becomes:
p
P (FP | T − 1000 ) = (1−p)
p+ 1000
(c) Assess Sara’s Strong Disbelief in Light of 1b
Sara’s intense skepticism stems from her emotional reaction to the number
1000, which she perceives as significant or "easy." However, when we analyze
the
situation using the formula from part 1b, it becomes clear that the posterior
probability of foul play is only considerable if the prior probability was already
substantial.
Thus, while Sara’s feelings of suspicion are understandable on an emotional
level, they do not hold up under statistical scrutiny. Her concerns are not
supported by
evidence that would indicate a higher likelihood of rigging in the raffle.
(d) Assess Mahra’s Argument in Light of 1b
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