A science article states: 'The present magnetic polarity has already lasted

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Earth, Climate & Energy Physics · The solid Earth: structure, fields and thermal history

The Solid Earth, Ch The geomagnetic field: potential, spectrum, induction and the palaeomagnetic record

Problem

A science article states: 'The present magnetic polarity has already lasted 0.78 Myr0.78\,\mathrm{Myr}, longer than the average interval length of 1/νr≈0.25 Myr1/\nu_r \approx 0.25\,\mathrm{Myr} (for a reversal rate νr=4 Myr−1\nu_r = 4\,\mathrm{Myr^{-1}}), so a reversal is now overdue and must happen soon.' ((a)) Treating reversals as a Poisson process with constant rate νr=4 Myr−1\nu_r = 4\,\mathrm{Myr^{-1}}, so that P(interval≥t)=e−νrtP(\text{interval}\ge t)=e^{-\nu_rt}, compute the probability that the present polarity, already 0.78 Myr0.78\,\mathrm{Myr} old, lasts at least a further 0.25 Myr0.25\,\mathrm{Myr}, that is P(interval≥1.03 Myr∣interval≥0.78 Myr)P(\text{interval}\ge1.03\,\mathrm{Myr}\mid\text{interval}\ge0.78\,\mathrm{Myr}). ((b)) Compare this with the probability that a new polarity interval, from its start, lasts at least 0.25 Myr0.25\,\mathrm{Myr}, and use the comparison to say whether the article's claim is justified.

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