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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Probability

NDA 2010 · Mathematics · Probability

An observed event B can occur after one of the three events A_1,...

An observed event B can occur after one of the three events A1,A2,A3\mathrm{A}_1, \mathrm{ A}_2, \mathrm{ A}_3. If P\mathrm{P}\left(\mathrm{A}_1\right)=\mathrm{P}(A2)=0.4,P(A3)=0.2\left(\mathrm{A}_2\right)=0.4, \mathrm{P}\left(\mathrm{A}_3\right)=0.2 and P(B∣A1)=0.25,P(B∣A2)=0.4\mathrm{P}\left(\mathrm{B} \mid \mathrm{A}_1\right)=0.25, \mathrm{P}\left(\mathrm{B} \mid \mathrm{A}_2\right)=0.4, P(B∣A3)=0.125\mathrm{P}\left(\mathrm{B} \mid \mathrm{A}_3\right)=0.125, what is the probability of A1\mathrm{A}_1 after observing B?

Options

  1. A1/31 / 3
  2. B6/196 / 19
  3. C20/5720 / 57
  4. D2/52 / 5

Correct answer

C. 20/5720 / 57

Step-by-step solution

P(A1B)=P(A1)×P(BA1)P(A1)×P(BA1)+P(A2)P(BA1)=0.4×0.250.4×0.25+0.4×0.4+0.2×0.125=410×25100410×25100+410×410+210×1251000=2057\begin{aligned} & \mathrm{P}\left(\frac{\mathrm{A}_1}{\mathrm{ B}}\right)=\frac{\mathrm{P}\left(\mathrm{A}_1\right) \times \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}_1}\right)}{\mathrm{P}\left(\mathrm{A}_1\right) \times \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}_1}\right)+\mathrm{P}\left(\mathrm{A}_2\right) \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}_1}\right)} \\ & =\frac{0.4 \times 0.25}{0.4 \times 0.25+0.4 \times 0.4+0.2 \times 0.125} \\ & =\frac{\frac{4}{10} \times \frac{25}{100}}{\frac{4}{10} \times \frac{25}{100}+\frac{4}{10} \times \frac{4}{10}+\frac{2}{10} \times \frac{125}{1000}}=\frac{20}{57}\end{aligned}
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