7. Let's make a guess for how long we would have to wait to see all the air atoms in a balloon move to one side - it is of course possible at least in principle. a) There are about 1022 atoms in a balloon. We found a single atom had a 50% chance of being on the left, so two atoms had a 50% times 50% chance of both being on the left, or 25%, so that three had a 12.5% chance of all three being on the left, and so on. If for any single atom there is a 50-50 chance of being on one side or the other, what is the probability that they will all be on the left side? (If you are having trouble calculating this you might want to recall that 10log(x)=x and that log(ab)=b log(a).) b) To change this probability into a waiting time, we need to divide it into the time it typically takes to change from one configuration into any other. A time that makes sense is the time it takes a single atom (moving at about the speed of sound, or 300 m/sec) to cross the 0.05 meters from one side of the balloon to the other. How long do we have to wait then to see this event with some reasonable probability? By way of comparison, the current age of the universe is roughly 10¹7 s.
7. Let's make a guess for how long we would have to wait to see all the air atoms in a balloon move to one side - it is of course possible at least in principle. a) There are about 1022 atoms in a balloon. We found a single atom had a 50% chance of being on the left, so two atoms had a 50% times 50% chance of both being on the left, or 25%, so that three had a 12.5% chance of all three being on the left, and so on. If for any single atom there is a 50-50 chance of being on one side or the other, what is the probability that they will all be on the left side? (If you are having trouble calculating this you might want to recall that 10log(x)=x and that log(ab)=b log(a).) b) To change this probability into a waiting time, we need to divide it into the time it typically takes to change from one configuration into any other. A time that makes sense is the time it takes a single atom (moving at about the speed of sound, or 300 m/sec) to cross the 0.05 meters from one side of the balloon to the other. How long do we have to wait then to see this event with some reasonable probability? By way of comparison, the current age of the universe is roughly 10¹7 s.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:7. Let's make a guess for how long we would have to wait to see all the air atoms in a balloon
move to one side - it is of course possible at least in principle.
a) There are about 1022 atoms in a balloon. We found a single atom had a 50% chance
of being on the left, so two atoms had a 50% times 50% chance of both being on
the left, or 25%, so that three had a 12.5% chance of all three being on the left, and
so on. If for any single atom there is a 50-50 chance of being on one side or the
other, what is the probability that they will all be on the left side? (If you are having
trouble calculating this you might want to recall that 10log(x)=x and that log(ab)=b
log(a).)
b) To change this probability into a waiting time, we need to divide it into the time it
typically takes to change from one configuration into any other. A time that makes
sense is the time it takes a single atom (moving at about the speed of sound, or 300
m/sec) to cross the 0.05 meters from one side of the balloon to the other. How long
do we have to wait then to see this event with some reasonable probability? By way
of comparison, the current age of the universe is roughly 1017 s.
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