EasyMathematicsHigh School (11–12)Statistics and Probability › Conditional Probability and Bayes' TheoremNumericalBoard ExamAPA-LevelIB
Box A holds 3 white and 7 black marbles…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Box A holds white and black marbles. Box B holds white and black marbles. A box is chosen at random, with box A chosen with probability and box B with probability . One marble is then drawn from the chosen box. Find the probability that the marble is white. Given that the marble is white, find the probability that it came from box A, and the probability that it came from box B.
Need a reminder? The key ideaShow
Bayes' theorem: , where is the sum of all the paths that end in . The top is one path, and the bottom is all of them.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The paths to white. Box A: . Box B: .
Step 2. White. .
Step 3. The posteriors. and .
Answer: , and .
Check. The two posteriors add up to . Box A was the less likely box before the draw, with , and a white marble makes it even less likely, because box A has fewer white marbles.
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