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Question:

A given sample of radium-226 has a half life of 4 days. What is the probability that a nucleus disintegrates after 2 half lives?

Attempt at solution:

After time $t$, the number of nuclei remaining will be $N = N_0e^{-kt}$ where $k$ is the decay constant, $k= \ln(2)/T_{1/2}$ , here $T_{1/2}$ is the half life, and $N_0$ is the initial number of nuclei. This implies $N$ nuclei have survived so far, hence , probability of survival $P= N/N_0$ , and hence probability of decay is $1-P$. Putting $t=2T_{1/2}$ in the equation, we get the required probability as $3/4$.

However, the given answer is $1/2$, the explanation provided was,

Disintegration of each nucleus is independent of any factor. Hence, each nuclei has same chance of disintegration.

I cannot understand this statement and also I don't understand what is wrong in my approach.

Any help is appreciated.

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    $\begingroup$ FWIW, Ra-226 actually has a half-life of 1600 years. No radium isotope has a half-life of 4 days, although Ra-224 has a half-life of 3.6319 days. $\endgroup$ – PM 2Ring May 9 at 10:05
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    $\begingroup$ I probably would have answered 1/4, because I understand "after" as "not before". $\endgroup$ – lvella May 9 at 12:19
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    $\begingroup$ I think they tried to phrase this question as conditional probability but failed miserably. $\endgroup$ – infinitezero May 9 at 13:23
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    $\begingroup$ @probably_someone Then the answer of 1/2 doesn't make sense. 1/2 is the probability it will decay within half life time from now, not instantaneously. $\endgroup$ – lvella May 9 at 13:30
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    $\begingroup$ Having been in the position of writing questions about radioactive decay, I've learned that (a) the question are hard to phrase unambiguously enough that my colleagues will all agree on what they mean and (b) even after you have done that a substantial fraction of students will misunderstand them because they have not understood the distinction made in defining the nomenclature in the first place. I got into the habit of always making these two part questions with the second part being "Explain your answer" so that students could earn partial credit for what they did understand. $\endgroup$ – dmckee May 9 at 15:17
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Your calculations and reasoning are correct, and the stated answer in your quote is wrong. Find a better textbook (if this is a practice problem) or report this error to your instructor (if it came from class materials).

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What an annoying exam question! As Emilio says, your reasoning & calculations are correct.

As your calculations indicate, if an isotope has a half-life of 4 days, then at any point in time a given nucleus of that isotope has a probability of .5 of decaying some time in the next 4 days, and a probability of .75 of decaying some time in the next 8 days, a probability of .875 of decaying some time in the next 12 days, etc.

So how can we make sense of the reason quoted at the end of your question?

Disintegration of each nucleus is independent of any factor. Hence, each nuclei has the same chance of disintegration.

There's a tiny chance (1/1024) that a nucleus of that isotope will survive 40 days, but that nucleus still has a probability of 0.5 of decaying some time in the subsequent 4 days.

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  • $\begingroup$ What you said about the nucleus that survived 40 days is correct, but it doesn't mean that is the correct answer to the question. Still doesn't make sense if that is the answer for 1 half life, while the question specifically asks about 2 half lives. $\endgroup$ – lvella May 9 at 12:37
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    $\begingroup$ @Ivella As I said, it's a terrible question. The nucleus has a probability of 3/4 of decaying within 2 half-lives. One way of reading "What is the probability that a nucleus disintegrates after 2 half lives?" Is that they are not asking about nuclei that decayed before 2 half-lives, therefore they're asking about the probability that a nucleus from the original sample decays in the time period commencing 2 half-lives after the start of the experiment, which is 1 - 3/4 = 1/4. $\endgroup$ – PM 2Ring May 9 at 13:01
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What does "What is the probability that a nucleus disintegrates after 2 half lives?" mean?

The question is horribly worded. I can come up with several interpretations, and there are probably more:

"What is the probability that a given nucleus does not decay until 2 half lives have passed?" Since by definition we know half of the nuclei will decay in the first half-life, and half of the remaining half in the second half-life, the probability is 1/4 as @Ivella says.

"What is the probability that a nucleus that didn't decay in the first two half-lives decays during the third?" the given answer of 1/2 is correct.

"What is the probability that a given nucleus that didn't decay during the first half-life also does not decay until 2 half lives have passed?" the given answer of 1/2 is correct.

"What is the probability that a nucleus that didn't decay in the first two half-lives decays after that time?" 100%, but it will literally take until the end of the universe for them all to actually decay.

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  • $\begingroup$ The interpretation issue is always a problem with these questions. You also left out What is the probability that the nucleus decay while the clock shows a time $2t_{1/2}$?" (Answer: $\Delta t/(4\tau)$ where $\tau = t_{1/2}/\ln 2$ is the lifetime and $\Delta t$ is the length of a clock tick; though that is a problem for more advanced student; at the intro level you could except zero as the probability of decay at exactly any given instant). I would discourage ever meaning your second and third possibilities. $\endgroup$ – dmckee May 9 at 15:25
  • $\begingroup$ @dmckee I agree that the 2nd and 3rd are not good interpretations, but they're the only ways I could get from the stated question to the official answer. And it's unfair for a teacher to expect a student to guess either of them rather than the first, which I believe to be the proper reading of the stated question. $\endgroup$ – Monty Harder May 9 at 15:30
  • $\begingroup$ Yes, it could take until the end of the universe for them all to actually decay. OTOH, Avogadro's number is slightly higher than $2^{79}$, so if we start with a mole, after 79 half-lives the odds aren't good that we have many undecayed atoms left. $\endgroup$ – PM 2Ring May 9 at 18:28
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Read literally, my answer might be "infinitessimal" and requiring of a definition of the minimum quantity of time sufficient to observe whether a nucleus has disintegrated or not. (The Heisenberg uncertainty principle might feature).

It is a quite appallingly badly worded question. The thought that it featured in an exam and that marks might have might have been awarded for the least obviously correct answer is sickening.

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The above question is from an Indian Joint Entrance Examination. The answer has been challenged and the key has been corrected to 3/4. So OP's approach is right (atleast one of the possibility correct ones).

NOTE: I also came across is question in an IITJEE Archive and asked my doubt to a faculty.

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  • $\begingroup$ Did they change it in the official key? $\endgroup$ – user226375 May 11 at 10:27
  • $\begingroup$ Yes, I do think so. $\endgroup$ – evamPUNdit May 11 at 16:05

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