# B. G. Cosmos, a scientist, believes that the probability is \frac{2}{5} that aliens from an advanced civilization on Planet X

B. G. Cosmos, a scientist, believes that the probability is $$\displaystyle{\frac{{{2}}}{{{5}}}}$$ that aliens from an advanced civilization on Planet X are trying to communicate with us by sending high-frequency signals to Earth. By using sophisticated equipment, Cosmos hopes to pick up these signals. The manufacturer of the equipment, Trekee, Inc., claims that if aliens are indeed sending signals, the probability that the equipment will detect them is $$\displaystyle{\frac{{{3}}}{{{5}}}}$$. However, if aliens are not sending signals, the probability that the equipment will seem to detect such signals is $$\frac{1}{10}$$. If the equipment detects signals, what is the probability that aliens are actually sending them?

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Tasneem Almond
Let A be the event of aliens sending signals and R be the event that the signals are received. If the equipment detects signals, the probability that aliens are actually sending them is:
$$\displaystyle{P}{\left({A}{\mid}{R}\right)}={\frac{{{P}{\left({A}\right)}{P}{\left({R}{\mid}{A}\right)}}}{{{P}{\left({A}\right)}{P}{\left({R}{\mid}{A}\right)}+{P}{\left({A}'\right)}{P}{\left({R}{\mid}{A}'\right)}}}}={\frac{{{\frac{{{2}}}{{{5}}}}{\frac{{{3}}}{{{5}}}}}}{{{\frac{{{2}}}{{{5}}}}{\frac{{{3}}}{{{5}}}}+{\frac{{{3}}}{{{5}}}}+{\frac{{{1}}}{{{10}}}}}}}={\frac{{{4}}}{{{5}}}}$$