Question

Consider the alpha particles emitted by a radioactive source during some time interval $t$. One can imagine this time interval to be subdivided into many small intervals of length $Delta t$. Since the alpha particles are emitted at random times, the probability of a radioactive disintegration occurring during any such time $Delta t$ is completely independent of whatever disintegrations occur at other times. Furthermore, $Delta t$ can be imagined to be chosen small enough so that the probability of more than one disintegration occurring in a time $Delta t$ is negligibly small. This means that there is some probability $p$ of one disintegration occurring during a time $Delta t$ (with $p ll 1$, since $Delta t$ was chosen small enough) and probability $1 - p$ of no disintegration occurring during this time. Each such time interval $Delta t$ can then be regarded as an independent trial, there being a total of $N = t/Delta t$ such trials during a time $t$. (a) Show that the probability $W(n)$ of $n$ disintegrations occurring in a time $t$ is given by a Poisson distribution. (b) Suppose that the strength of the radioactive source is such that the mean number of disintegrations per minute is 24. What is the probability of obtaining $n$ counts in a time interval of 10 seconds? Obtain numerical values for all integral values of $n$ from 0 to 8.

          Consider the alpha particles emitted by a radioactive source during some time interval $t$. One can imagine this time interval to be subdivided into many small intervals of length $Delta t$. Since the alpha particles are emitted at random times, the probability of a radioactive disintegration occurring during any such time $Delta t$ is completely independent of whatever disintegrations occur at other times. Furthermore, $Delta t$ can be imagined to be chosen small enough so that the probability of more than one disintegration occurring in a time $Delta t$ is negligibly small. This means that there is some probability $p$ of one disintegration occurring during a time $Delta t$ (with $p ll 1$, since $Delta t$ was chosen small enough) and probability $1 - p$ of no disintegration occurring during this time. Each such time interval $Delta t$ can then be regarded as an independent trial, there being a total of $N = t/Delta t$ such trials during a time $t$.
(a) Show that the probability $W(n)$ of $n$ disintegrations occurring in a time $t$ is given by a Poisson distribution.
(b) Suppose that the strength of the radioactive source is such that the mean number of disintegrations per minute is 24. What is the probability of obtaining $n$ counts in a time interval of 10 seconds? Obtain numerical values for all integral values of $n$ from 0 to 8.

Consider the alpha particles emitted by a radioactive source during some time interval t. One can imagine this time interval to be subdivided into many small intervals of length Delta t. Since the alpha particles are emitted at random times, the probability of a radioactive disintegration occurring during any such time Delta t is completely independent of whatever disintegrations occur at other times. Furthermore, Delta t can be imagined to be chosen small enough so that the probability of more than one disintegration occurring in a time Delta t is negligibly small. This means that there is some probability p of one disintegration occurring during a time Delta t (with p ll 1, since Delta t was chosen small enough) and probability 1 - p of no disintegration occurring during this time. Each such time interval Delta t can then be regarded as an independent trial, there being a total of N = t/Delta t such trials during a time t.
(a) Show that the probability W(n) of n disintegrations occurring in a time t is given by a Poisson distribution.
(b) Suppose that the strength of the radioactive source is such that the mean number of disintegrations per minute is 24. What is the probability of obtaining n counts in a time interval of 10 seconds? Obtain numerical values for all integral values of n from 0 to 8.

Added by Kelsey C.

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