Frank Morgan's Math Chat


Chance of Bank Balance $0.00

Old Challenge. My colleague Dick De Veaux reports that upon returning from a three-week trip, he found his checking account balance at exactly $0.00. Without further information, how would you estimate the probability of such an event?

Answer (Joseph DeVincentis). "As far as the checking account balance, it depends to a huge degree on the person's spending habits, how much money they normally have in their checking account, etc. For me, now, the answer would be 0. I never let my account balance drop so low. This wasn't always the case, though; I remember one time in college having $4 and some odd cents in my checking account, and $1 in my wallet. But you asked how I would estimate it without further information, so...

To make a practical answer, let's assume that 0 is a reasonable balance for this person's account after the three week trip. Airfare and hotel costs are probably planned ahead of time, and for the purposes of this problem can be considered fixed. However, the costs of meals and other incidental expenses can vary. We don't know how many people were with him either, and a lot of these expenses are roughly proportional to the number of people, so I'm going to assume it was him and his wife. If they eat cheaply and don't have a lot of other expenses, they might manage to spend only $30 a day. At the other extreme, if they eat lavishly and spend a lot on shopping or entertainment, they might spend $200 or more per day. However, for any specific person, the variation is going to be less than this, based on that person's budget. I'm going to assume a range of possible variation of about $50 per day, considering that he's there for three weeks, and also that the extremes will have some tendency to average out. That makes a range of about $1000 of possible variation over the entire trip. If all of this was charged against his checking account, and not put on credit cards, etc., then this makes for 100000 possible different to-the-penny amounts, and a chance of about 1 in 100000 that any specific sum was the final balance in his checking account. [Not necessarily, since if the balance goes below 0, it might not get a chance to bounce back.] There is of course more likelihood of numbers in the middle of the range than ones on the outside, but we don't know where 0 falls in the range! However, given that banks tend to charge severe penalties for overdrawing your account, most people try to avoid this situation, so I will assume that a balance of 0 falls on the side of heavier spending, so it falls in a range of values that are less likely than more moderate values, and so I'll throw in a factor of 2 for that and call it a 1 in 200000 chance."

Math Chat notes that with almost 300 million people in the country, this probably happens all the time.

New Challenge. Suppose that in a certain US state 12 million potential voters favor Bush, 10 million favor Gore, and 5 million favor Nader with Gore as second choice. The Nader supporters want to be sure than Gore wins, but they would like Nader to get lots of votes, too. If you could send one common short message of friendly advice to the Nader supporters, what would you tell them to do?

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