Question

In 2019, Pittsburgh Steelers starting quarterback Ben Roethlisberger suffered a season-ending elbow injury in Week 2. The team struggled to find a quarterback until Week 12, when undrafted rookie Devlin “Duck” Hodges entered a game against the Cincinnati Bengals at halftime and led the Steelers to a surprising comeback victory. His performance earned him the starting job for the rest of the season and engendered a flurry of media speculation about his potential as a future quarterback for the Steelers.

Assume quarterbacks can either be good, average, or bad. Because Hodges was undrafted, it’s fair to say that coaches and fans had low expectations for him. Assume the initial probability distribution for Hodges was:

P(Good) =0.1

P(Average)=0.2

P(Bad)= 0.7

Also assume quarterbacks can either have strong performances or weak performances (there are only two types of performances, strong or weak). The chance of a good quarterback having a strong performance is 80%, the chance of an average QB having a strong performance is 50%, and the chance of a bad QB having a strong performance is 30%. We have:

P(S | G) =0.8

P(S | A)=0.5

P(S | B)= 0.3

From ESPN.com, Hodges turned in the following performances during the last 6 weeks of the season:

Date	Opponent	Hodges Passer Rating	Evaluation*
24-Nov	Bengals	115	Strong
1-Dec	Browns	95.7	Strong
8-Dec	Cardinals	117.5	Strong
15-Dec	Bills	43.9	Weak
22-Dec	Jets	37	Weak
29-Dec	Ravens	47.9	Weak
*the average passer rating is around 88

Use Bayes’s Rule to fill in the following table with the week-by-week probabilities of Duck being good, average, or bad.

Consecutive Wins	Prior to the Season	Game 1	Game 2	Game 3	Game 4	Game 5	Game 6
Chances of Being Good	10%
Chances of Being Average	20%
Chances of being Bad	70%

Answer 1

According to Bayes Rules:

The same formula will be for P(A/S) and P(B/S) where G in numerator will be changes by A and B.

The same formula will be for P(G/W), P(A/W) and P(B/W) where S will be changes by W.

for week 1. we use prior probability of duck being good average and bad. for week 2 we will use the week 1 probability in bayes rules. similar steps will be done for all the week.

Note: first three week have strong performance so we calculated P(G/S) P(B/S) and P(A/S) but for next three we will use P(G/W) P(B/W) and P(A/W)

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working note:

week 1: first entry

week 5: first entry

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