Probability and statistics

In a night exercise three platoons, A,B,C are required to cross a stretch of ground in one hour without making so much noise that they are detected by the adjudicating officer. The platoons draw lots for the starting line times of the mid-night, 1.00 and 2.00, the best platoons, A, has the probability 3/4 of crossing sufficiently quietly while B,C have probabilities 1/2 and 1/4 of doing so. If the adjudicating officer hears no noise in the first hour, but some noise in each of the next two hours, what is the probability that the platoons crossed in alphabetical order?

Re: Probability and statistics

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**samuel2012** In a night exercise three platoons, A,B,C are required to cross a stretch of ground in one hour without making so much noise that they are detected by the adjudicating officer. The platoons draw lots for the starting line times of the mid-night, 1.00 and 2.00, the best platoons, A, has the probability 3/4 of crossing sufficiently quietly while B,C have probabilities 1/2 and 1/4 of doing so. If the adjudicating officer hears no noise in the first hour, but some noise in each of the next two hours, what is the probability that the platoons crossed in alphabetical order?

We must make an assumption not given, namely that the crossings are independent.

We are given that two of the crossings are 'noisy'.

Let's use $\displaystyle A$ to mean that platoons A has a quite crossing, and $\displaystyle \overline{B}$ to mean that platoons B has a noisy crossing.

Here is what we are given $\displaystyle A{\overline{B}\,\overline{C}\cup A{\overline{C}\,\overline{B}$$\displaystyle \cup B{\overline{A}\,\overline{C}\cup B{\overline{C}\,\overline{A}$$\displaystyle \cup C{\overline{A}\,\overline{B}\cup C{\overline{B}\,\overline{A}$

Now $\displaystyle \mathcal{P}(A{\overline{B}\,\overline{C})=(.75)(.5 )(.75)$

Now your turn.