Problem 1. Raffle
(#Probability, #CompTools)
On a recent visit to Global Village in Dubai, three ZU students, Sara, Mahra, and
Ahmad, attended a raffle event where there was a box with 1000 tickets, each with a
unique number starting from 1 to 1000. A ticket was randomly taken out and the
winning number was 1000.
This led to the following discussion among Sara, Mahra, and Ahmad.
Sara believes that it's very unlikely that such an easy number, 1000, was
randomly drawn,
and she suspects that the drawing must be rigged, hence a foul play.
Mahra argues that while 1000 is the largest number but any number between 1
and 1000 is
equally unlikely so no reason to suspect foul play.
Ahmad, after listening to Mahra, got convinced and argued that there was a
0.1% chance of
foul play.
Question 1 of 18
1a. Who do you think is right or wrong and why?
• Right: Mahra is correct. In a fair random drawing, every number has an equal
probability of being selected, including the number 1000. Therefore, there is no
reason to suspect foul play simply because 1000 was drawn.
• Wrong: Sara is wrong to assume foul play without any evidence. Ahmad's conclusion
about a 0.1% chance of foul play is unfounded without further context or data to
support such a claim.
,Question 2 of 18 1b.
Suppose if p represents the prior probability that the raffle is rigged then calculate the
posterior probability of foul play.
To calculate the posterior probability of foul play (rigging of the raffle), we can use Bayes'
Theorem, which is expressed as follows:
P( B ∣ A) ⋅ P ( A )
P( A ∣ B)=
P ( B)
Where:
P(A∣B) is the posterior probability of the event A (foul play) given evidence B
(drawing of number 1000).
P(B∣A) is the probability of the evidence given that the event A is true.
P(A) is the prior probability of the event A (foul play).
P(B) is the total probability of the evidence.
1. Define Variables:
o Let p=P ( A ): the prior probability that the raffle is rigged.
o Let P( B ∣ A)=1: the probability of drawing 1000 if the raffle is rigged
(assuming the rigging guarantees 1000 is drawn).
1
o Let P( B ∣¬ A)= : the probability of drawing 1000 if the raffle is not
1000
rigged (where every ticket has an equal chance).
2. Calculate P(B):
o Using the law of total probability:
P( B)=P( B ∣ A)⋅ P( A)+ P(B ∣¬ A)⋅ P(¬ A)
1
P ( B )=1⋅ p+ ⋅(1− p)
1000
1− p
P ( B )= p+
1000
3. Substitute into Bayes' Theorem:
P( B ∣ A) ⋅ P ( A )
P( A ∣ B)=
P ( B)
p
P ( A ∣B )=
1− p
p+
1000
, 4. Final Expression for Posterior Probability:
p
P( A ∣ B)=
1− p
p+
1000
Question 3 of 18 1c.
Assess Sara’s strong disbelief in the raffle in the light of your answers to part 1b.
Rational Disbelief:
If Sara’s prior probability reflects a reasonable assessment of the situation based on
her experiences or evidence (e.g., past instances of rigging), her disbelief might be
justified. However, if her belief is based solely on the outcome (drawing 1000), it
lacks a statistical foundation.
Since Mahra pointed out that all numbers are equally likely, this suggests that Sara's
suspicion might be more emotional or intuitive rather than grounded in probability.
Conclusion:
While Sara's disbelief is understandable from an intuitive standpoint, the analysis
indicates that a random draw can indeed result in any number, including 1000,
without suggesting foul play. Thus, if her disbelief does not take into account the
equal probability of all outcomes, it might be excessive and not supported by the
mathematical reasoning established through Bayes' Theorem.
Question 4 of 18 1d.
Assess Mahra’s argument in the light of your answers to part 1b.
Mahra's argument centers around the idea that since all numbers in a fair raffle are equally
likely to be drawn, there is no reason to suspect foul play when 1000 is drawn. Here’s the
assessment based on the analysis from part 1b:
1. Correct Understanding of Probability:
o Mahra correctly points out that in a fair raffle with 1000 tickets, every ticket,
including the number 1000, has a probability of 1/1000 of being drawn. This
(#Probability, #CompTools)
On a recent visit to Global Village in Dubai, three ZU students, Sara, Mahra, and
Ahmad, attended a raffle event where there was a box with 1000 tickets, each with a
unique number starting from 1 to 1000. A ticket was randomly taken out and the
winning number was 1000.
This led to the following discussion among Sara, Mahra, and Ahmad.
Sara believes that it's very unlikely that such an easy number, 1000, was
randomly drawn,
and she suspects that the drawing must be rigged, hence a foul play.
Mahra argues that while 1000 is the largest number but any number between 1
and 1000 is
equally unlikely so no reason to suspect foul play.
Ahmad, after listening to Mahra, got convinced and argued that there was a
0.1% chance of
foul play.
Question 1 of 18
1a. Who do you think is right or wrong and why?
• Right: Mahra is correct. In a fair random drawing, every number has an equal
probability of being selected, including the number 1000. Therefore, there is no
reason to suspect foul play simply because 1000 was drawn.
• Wrong: Sara is wrong to assume foul play without any evidence. Ahmad's conclusion
about a 0.1% chance of foul play is unfounded without further context or data to
support such a claim.
,Question 2 of 18 1b.
Suppose if p represents the prior probability that the raffle is rigged then calculate the
posterior probability of foul play.
To calculate the posterior probability of foul play (rigging of the raffle), we can use Bayes'
Theorem, which is expressed as follows:
P( B ∣ A) ⋅ P ( A )
P( A ∣ B)=
P ( B)
Where:
P(A∣B) is the posterior probability of the event A (foul play) given evidence B
(drawing of number 1000).
P(B∣A) is the probability of the evidence given that the event A is true.
P(A) is the prior probability of the event A (foul play).
P(B) is the total probability of the evidence.
1. Define Variables:
o Let p=P ( A ): the prior probability that the raffle is rigged.
o Let P( B ∣ A)=1: the probability of drawing 1000 if the raffle is rigged
(assuming the rigging guarantees 1000 is drawn).
1
o Let P( B ∣¬ A)= : the probability of drawing 1000 if the raffle is not
1000
rigged (where every ticket has an equal chance).
2. Calculate P(B):
o Using the law of total probability:
P( B)=P( B ∣ A)⋅ P( A)+ P(B ∣¬ A)⋅ P(¬ A)
1
P ( B )=1⋅ p+ ⋅(1− p)
1000
1− p
P ( B )= p+
1000
3. Substitute into Bayes' Theorem:
P( B ∣ A) ⋅ P ( A )
P( A ∣ B)=
P ( B)
p
P ( A ∣B )=
1− p
p+
1000
, 4. Final Expression for Posterior Probability:
p
P( A ∣ B)=
1− p
p+
1000
Question 3 of 18 1c.
Assess Sara’s strong disbelief in the raffle in the light of your answers to part 1b.
Rational Disbelief:
If Sara’s prior probability reflects a reasonable assessment of the situation based on
her experiences or evidence (e.g., past instances of rigging), her disbelief might be
justified. However, if her belief is based solely on the outcome (drawing 1000), it
lacks a statistical foundation.
Since Mahra pointed out that all numbers are equally likely, this suggests that Sara's
suspicion might be more emotional or intuitive rather than grounded in probability.
Conclusion:
While Sara's disbelief is understandable from an intuitive standpoint, the analysis
indicates that a random draw can indeed result in any number, including 1000,
without suggesting foul play. Thus, if her disbelief does not take into account the
equal probability of all outcomes, it might be excessive and not supported by the
mathematical reasoning established through Bayes' Theorem.
Question 4 of 18 1d.
Assess Mahra’s argument in the light of your answers to part 1b.
Mahra's argument centers around the idea that since all numbers in a fair raffle are equally
likely to be drawn, there is no reason to suspect foul play when 1000 is drawn. Here’s the
assessment based on the analysis from part 1b:
1. Correct Understanding of Probability:
o Mahra correctly points out that in a fair raffle with 1000 tickets, every ticket,
including the number 1000, has a probability of 1/1000 of being drawn. This
