A group of investigative journalists are writing a carefully researched story that hinges in part on the whereabouts of a person x. As part of this, they have obtained a metro card that they think belonged to x. The metro card was a widely-used type of bimonthly pass that allowed unlimited travel on the metro during weekdays in the two-month period covering August and September 2019. They're trying to decide how confident to be that this metro card in fact belonged to x. In principle, the metro card could have belonged to anyone who bought such a card in August/September 2019. But the journalists have collected the following interesting evidence. The main thing they've done is to recruit a friend of theirs who can read the digital encoding on metro cards, and from this they've discovered that the metro card was used every weekday of August and September 2019 except for August 6, 7, 13, 20, and 28; it was not used on these five weekdays. (There were a total of 43 weekdays in August and September 2019. Also, note that the card was used every weekday in September.) They then checked with the company that employed x in August and September 2019, and they learned that x did take the metro to work, and that x was at work every weekday those two months except for five days: August 6, 7, 13, 20, and 28. This seems fairly compelling, except for the fact that a million people in the city possessed a metro card like this for the two-month period covering August and September 2019, and on any given weekday, 80% of these metro cards were used. So perhaps other people had this same pattern of riding the metro during those two months. The journalists don't have access to any of the other metro cards that were used those two months, but they'd nevertheless like to estimate the probability that the usage pattern on the metro card they have uniquely identifies x. To do this, they make the following assumptions to serve as a stylized representation of the facts they know:

They model the population of one million metro-card users by assuming that each of them independently chose to use their metro card each day with probability .8 (and not to use it with probability .2).

They conclude, based on their communications with x's employer, that x used their metro card every weekday of August and September 2019 except for August 6, 7, 13, 20, and 28, and didn't use their card on those five days. The basic question is then: how likely is it that there was only a single metro card that was used on exactly this set of days? If it's very likely, then they can conclude based on their assumptions that there's a high probability they have x's metrocard; x has been uniquely identified. Here is how they go about answering this question.

(3.1) (4 points) Let's say that some other metro-card holder (other than x) matches x if they used their metro card on every weekday of August and September 2019 other than August 6, 7, 13, 20, and 28, and did not use it on these five weekdays. Pick some specific metro-card holder y who is not x. What is the probability that y matches x, given the assumptions being made? Briefly explain your answer.

(3.2) (4 points) What is the probability that there is no metro-card holder (other than x) who matches x, given the assumptions being made? Briefly explain your answer. Note that this probability is the probability that the usage pattern on the metro card the journalists have uniquely identifies x, given their assumptions (since it would mean that the only metro card with x's pattern of usage is the one that is in the journalists' possession).

(3.3) (8 points) Finally, you meet up with the journalists, and you'd like to convince them that they should be more careful before making any claims with confidence. In particular, name at least two assumptions they're making that could affect the probability they're arriving at. For each, specify the assumption, why it might not hold, and in general terms, without computing specific numbers why it might affect their calculation of the conditional probability. (Note that the earlier parts of this question asked for the probabilities in terms of the model as the journalists had set it up; it's only this part that asks you to call some of the assumptions of the model into question.)