Miss Marple wondered if Agatha might be guilty of murdering her father. She certainly had a financial motive to do so! Gordon the Gardener stomped into Miss Marple’s room and claimed that he had seen Agatha was outside in the garden adjacent to the main house at around the time the murder occurred, and that Agatha therefore could not be the murderer. Miss Marple, ever alert, observed his nervousness and unwillingness to look her in the eye, and felt uncertain about this story. Gordon left, and she pondered on his truthfulness, or otherwise.
If Agatha truly were in the garden then it might seem she couldn’t be the murderer... except that, as an accomplished knife thrower, she could have struck Sir Humphrey with a well- aimed throw straight through the open dining room window! Such a throw seemed possible but unlikely, Miss Marple mused, and she assigned a probability of 0.1 to Agatha being the murderer if she had actually been outside.
However if Agatha had been inside after all, and Gordon had been lying, then Agatha seemed a much more likely suspect. Miss Marple assigned a probability of 0.8 of Agatha being the murderer had she actually been inside the house at the time.
After some more deep thought, and refreshed by nice cup of tea, Miss Marple determined her probability that Agatha was the murderer to be 0.6. What probability did Miss Marple assign to the truth of Gordon the Gardener’s story about Agatha being outside in the garden?
I can't seem to solve it. I tried to create a notation but I'm not sure if it's correct.
$$P(\text{murder}) = 0.6$$ $$P(\text{murder} | \text{outside}) = 0.1$$ $$P(\text{murder} | \text{inside}) = 0/8$$ $$P(\text{innocent}) = 1 - 0.6 = 0.4$$
Do I use total probability? Or Bayes' theorem?
Total probability = P(outside) = P(outside|murder) P(murder) + P(outside|innocent) P(innocent)
But I cannot find most of the probability? If I use Bayes' theorem, the same thing happens.
P(outside | murder) = P(murder | outside) P (outside) / P(murder) and yet I don't have P(outside).
Miss Marple next turned her little grey cells to The Butler. Simply because he was a butler, Miss Marple thought it was a 50/50 chance that he was the murderer; it was just as likely that he was the murderer as that he wasn’t. But then Harriet the Housekeeper, who had delivered the tea, volunteered that as she rushed past the kitchen on the way to the adjacent dining room to investigate all the noise, she had seen The Butler vigorously washing his hands... (Harriet wondered aloud if he might have been washing blood off his hands?)
Miss Marple asked for more tea as she thought about this new information. She trusted Harriet’s account of what she had seen, but The Butler was a known cleanliness fanatic — while vigorous handwashing at that late hour might seem a little odd, she thought that if he had not been murdering Sir Humphrey next door there would still be a 25% chance that he would be washing his hands like that anyway. On the other hand, if he had just been murdering his boss, it was very likely, say 95%, that he would have been cleaning his hands as if his life depended on it (which it just might!).
How does Miss Marple update her probability that The Butler is the murderer?
$$P(\text{murder}) = 0.5$$ $$P(\text{murder} | \text{wash}) = 0.95$$ $$P(\text{innocent} | \text{wash}) = 0.25$$
I'm not quite sure what is being asked here.
For the first part, you can just work with the conditional probabilities you know. Apply total probability to $P(\text{murder})$:
$$P(\text{murder}) = P(\text{murder} \mid \text{outside}) P(\text{outside}) + P(\text{murder} \mid \text{inside}) P(\text{inside}).$$
And of course $P(\text{inside}) = 1 - P(\text{outside}),$ so you can write an equation with just one unknown variable.
For the second part, what the problem statement actually says is that $P(\text{wash} \mid \text{murder}) = 0.95$ and $P(\text{wash} \mid \text{innocent}) = 0.25.$