Find the probability that a muon is passing through your body this instant

[JFuld]

Homework Statement

the flux of muons on the Earth's surface is about 100 muons per square meter per second. Estimate the probability that a muon is passing
through your body this instant to within a factor of three

attempt:

let F = flux of muons on Earth's surface = 100/m^2/s

let the average person occupy an area A on the Earth's surface.

then F*A =# of muons penetrating person every second

I don't really know where to go from here. if there are N "instants" in a second then FA/N gives the average # of muons passing through you in an instant?

I feel i am going about this wrong, any help is appreciated.

 

[tiny-tim]

Hi JFuld!

Estimate the probability that a muon is passing
through your body this instant to within a factor of three
…
… if there are N "instants" in a second

no, an instant is zero time,

it means how many are likely to be inside your body at any fixed time

(so you'll need an estimate of their speed )

 

[JFuld]

ahh thank you, I knew I had no idea what I was doing haha. Well here is what I did in sight of your comment:

Im still using F*A =# of muons penetrating person every second, and I am letting v=vertical velocity of muons. Also, I am picturing the incident muons as a steady stream.

v*(1 second) = d, the distance traveled by muons in one second. So a vertical segment of length d contains 100 muons/m^2.

and A*d corresponds to the # of muons contained in segment d over an area A.

then taking A to equal .25m^2, the volume d*A contains 25 muons at an instant.

and 25/d =# muons per area A, then h*25/d = N = the # of muons occupying a person at an instant

also, my estimate for the muon velocity is: 1*10^8 m/s < v < c

then the range for d is: 1*10^8 m < d < 3*10^8 m

this give the range for N: 4.5*10^-7 < N < 1.5*10^-7 (i aproximate h to be 1.8 m)

 

[tiny-tim]

looks good!

 

[paranormal]

I would approach this problem by first acknowledging that the probability of a muon passing through my body at any given instant is extremely low. However, since the flux of muons on Earth's surface is relatively high at 100 muons per square meter per second, there is still a chance that a muon could pass through my body in any given instant.

To estimate the probability, we can use the concept of Poisson distribution, which is commonly used in physics to calculate the probability of rare events. In this case, we can consider each instant as a trial and the number of muons passing through my body as the number of successes.

Using the given flux of 100 muons per square meter per second, we can calculate the expected number of muons passing through my body in one second as:

Expected number of muons = 100 muons/m^2/s * A * 1 second

Where A is the average area occupied by a person on Earth's surface.

Since we are asked to estimate the probability to within a factor of three, we can use the range of values from 1/3 to 3 times the expected number of muons as the possible range of successes.

Therefore, the probability of a muon passing through my body in one instant can be estimated as:

Probability = (1/3) * (100 muons/m^2/s * A * 1 second) to (3) * (100 muons/m^2/s * A * 1 second)

= (33.33 muons * A) to (300 muons * A)

This means that for every square meter of area occupied by a person, the probability of a muon passing through their body in one instant is between 33.33 and 300 muons. This is a very low probability, but it is still possible.

It is important to note that this is just an estimation and the actual probability may vary depending on factors such as the specific area occupied by the person, the duration of the instant, and fluctuations in the muon flux. But using the concept of Poisson distribution, we can get a rough estimate of the probability of a muon passing through our body in one instant.