A mathematical treatment of some biological problems / Shinkishi Hatai.

  • Shinkishi Hatai
Date:
1910
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Licence: In copyright

Credit: A mathematical treatment of some biological problems / Shinkishi Hatai. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

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    BIOLOGICAL PROBLEMS. SHINKISHI HATAI, Associate in Neurology, The Wistar Institute of Anatomy. Dr. King’s work (’07, ’09), “On Studies on Sex-determination in Amphibians,” suggests an interesting problem which may be put in the following form : A jar contains a large number of male and female tadpoles, the proportion of each being unknown : if on picking out m -f n tad- poles, m are found to be males and 71 females, to find the prob- ability that the ratio of the number of either sex to the entire lot lies between given limits. It will be seen that problems of this nature occur frequently in biological investigations and that it is of importance to have some method for determining the accuracy of the observed pro- portions. This can be done by means of the formula given below. As the development of the formula is somewhat complicated I shall present the entire process of the mathematical treatment of the solutions based on the theorem of Bayes, together with one application. Although such an elementary exposition of the sub- ject will be superfluous for one who is familiar with the theory of probabilities nevertheless for others it may be helpful. If p denote the probability that an event will happen, then (1 — p) is the probability that the event will fail. If the proba- bility that the event will fail on any single trial is (1 —p), the probability that it will fail every time is (1 —p)n. The proba- bility that it will happen on the first trial and fail on the succeed- ing n — I trial is/(i — p)n~l. But the event is just as likely to happen on the second, third, etc., trials as on the first. Hence the probability that the event will happen just once in the n trials is np{ I Continuing this process, we can easily see that the probability that it will happen m times in m -f n trials is
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