Probability exercise I found difficult An obervatory has 4 radio locators, each with the following probabilities of finding an object: first locator - 0.86, second - 0.89, third - 0.9 and the fourth - 0.95. a) One of the locators is randomly switched on. What is the probability of finding an object? b) An object is found after switching on one of the locators. What is the probability that it was the third locator that was switched on? I am a beginner with statistics and found this exercise very hard.

For $i=1,2,3,4$ let $p_i=P(\text{found obj.}\mid \text{locator $i$ is on})$ be as given in the question. Also, $P(\text{locator $i$ is on})=1/4$ for all $i$.

a) $P(\text{found obj.}) = \sum_{i=1}^4 P(\text{locator $i$ is on, found obj.}) = \sum_{i=1}^4 P(\text{found obj.} | \text{locator $i$ is on})P(\text{locator $i$ is on}) = \frac{1}{4} \sum_{i=1}^4 p_i$.

b) Use Bayes's rule on $P(\text{locator $3$ is on} \mid \text{found obj.})$.