Statistics and probability (a) Two sets of n independent trials are performed, independently of each other, and each trialresults in either success or failure, the probability of success being p1 in the first set of trials andP2 in the second set. Show that the probability P of obtaining x1 successes in the first set andx2 successes in the second set is given byP = Kpipi (1 – p1)"*(1 – p2)"*,where K depends only on n, x1 and x2. If p1 =p and p2 = p?, find an expression for log Pand show that, for given values of n, x, and x2, log P has a maximum value when p is suchthat(x1 + 2x2) – (n - x)p – 3np² = 0.(b) An insect breeding experiment was conducted in two sections, in each of which 100 insectsof a particular species were raised. In the first section a proportion p was expected to have acertain colour variation and in the second section the proportion with this colour variation wasexpected to be p², but the value of p was not known. In the event there were 22 insects in thefirst section, and 7 in the second section, which possessed the colour variation. Find the valueof p which maximises the probability of this result.

Answered: Image /qna-images/answer/110f19f4-7c8d-4904-9ea2-b3dec2a1117b.jpg

(a) Two sets of n independent trials are performed, independently of each other, and each trial results in either success or failure, the probability of success being p1 in the first set of trials and P2 in the second set. Show that the probability P of obtaining x1 successes in the first set and x2 successes in the second set is given by P = Kpi'p? (1 – p1)" *(1 - p2)"*, where K depends only on n, x1 and x2. If p1 =p and p2 = p?, find an expression for log P and show that, for given values of n, x1 and x2, log P has a maximum value when p is such that (x, + 2x2) – (n – x1)p - 3np? = 0. (b) An insect breeding experiment was conducted in two sections, in each of which 100 insects of a particular species were raised. In the first section a proportion p was expected to have a certain colour variation and in the second section the proportion with this colour variation was expected to be p², but the value of p was not known. In the event there were 22 insects in the first section, and 7 in the second section, which possessed the colour variation. Find the value of p which maximises the probability of this result.

(a) Two sets of n independent trials are performed, independently of each other, and each trial results in either success or failure, the probability of success being p1 in the first set of trials and P2 in the second set. Show that the probability P of obtaining x1 successes in the first set and x2 successes in the second set is given by P = Kpi'p? (1 – p1)" (1 - p2)", where K depends only on n, x1 and x2. If p1 =p and p2 = p?, find an expression for log P and show that, for given values of n, x1 and x2, log P has a maximum value when p is such that (x, + 2x2) – (n – x1)p - 3np? = 0. (b) An insect breeding experiment was conducted in two sections, in each of which 100 insects of a particular species were raised. In the first section a proportion p was expected to have a certain colour variation and in the second section the proportion with this colour variation was expected to be p², but the value of p was not known. In the event there were 22 insects in the first section, and 7 in the second section, which possessed the colour variation. Find the value of p which maximises the probability of this result.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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