1. Planet-I-Pay is a digital payment service at IIM Indore that offers guaranteed rewards for every eligible transaction. Suppose there are 25 unique rewards offered by the system, and it gives out a reward randomly from the pool after every eligible transaction (so one may get the same reward multiple times). Argue with justification that the number of eligible transactions you need to make to get all the 25 rewards is a random variable. Find the expected number of eligible transactions you need to make to get all the 25 rewards. Suppose you have already received 10 unique rewards. Find the expected number of eligible transactions you would now need to make to get the remaining 15 rewards.
[10 points]
2. The Covid-19 pandemic had changed the way our daily lives operate. During the un- locking phase, when gymnasiums opened up again, some new norms were put into place. Suppose there are 5 treadmills in a gym, placed side-by-side. New social distancing norms mandated that no two contiguous treadmills should be occupied at the same time. In- dividuals arrive at the gym when it opens in the morning and queue up. One by one, each individual randomly picks a treadmill available for use, until no more treadmills are available (respecting the new mandate). Find the expected number of individuals using the treadmills at this point.
[10 points]
3. Look into the report Technical Briefing 20, published on August, 2021 by Public Health England regarding current SARS-COV2 variants of concern and variants under investigation in England. Specifically, look into Table 5 on pages 18-19. You will see that there were 402 Covid deaths out of 47,008 fully vaccinated individuals, leading to a Case Fatality Rate (CFR) of 0.86%, whereas there were 253 Covid deaths out of 151,054 unvaccinated individuals, leading to a CFR of 0.17%. Here CFR refers to the proportion of confirmed Covid cases (all Delta variants) that resulted in death.
Thus, we see that the CFR for vaccinated is more than 5-fold higher than for unvaccinated.
How would you argue with an Anti-vaxx campaigner on this? Present a brief write-up with your arguments and supporting theoretical/empirical evidences.
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