Bayes Rule

[mathstats]

John and Hillary have each thought about murdering the president. John has put rat poison in the cherry cobbler; Hillary, unaware, has put cyanide in the chocolate pie. Given the amounts likely to be eaten, the probability of the rat poison being fatal is .6; the cyanide, .75. Based on other dinners, we can assume that he has a 50% chance of asking for the cherry cobbler, 45% chance with the chocolate pie, and a 5% chance of not eating dessert. No sooner are the dishes cleaned and the president drops dead. Who should be considered the prime suspect?

1. Find the probability that he dies by any method
2. Find the probability that he dies by each separate method given that he dies.
3. Determine which is the most likely cause of death?

 

[Dr.Peterson]

What a sick problem!

Both are guilty of attempted murder; one is just slightly more likely to be the actual cause of death (though, to be precise, the problem omits the probability of other causes, such as heart attack, so we can't really answer).

Now, if you want us to help you solve the problem (before your teacher is arrested), you'll have to show us your own thoughts, so we can know where you need help, and what knowledge you have that we can start with. Can you, for example, state Bayes' Rule?

 

[firemath]

Oh dear. What a lovely opening line!?

John and Hillary have each thought about murdering the president.

 

[HallsofIvy]

I have never been good at applying "Baye's rule". Here is how I would do this:
(I also need to state my distaste for the wording of this problem! I almost decided not to respond but I cannot resist this type of problem.)

Imagine 10000 repetitions of this scenario, in 50%, 5000, the person eats the peach cobbler, in 45%, 4500, the chocolate pie, and in 5%, 500, no dessert. In the 5000 cases in which he at the peach cobbler, 0.6 of the time, 3000, he dies.. In the 4500 cases in which he ate the chocolate pie, 0.75 of the time, 3375, he dies. In the 500 times when he eats no desert, of course, he does not die.

So of the 10000 possible scenarios, he dies in 3000+ 3375= 6375. The probability he dies is 6375/10000= 0.6375 of 63.75%.

Of the 6375 times he dies, 3000 of then were from the rat poison administered by John. That is 3000/6375= 0.47, 47%. 3375 were from the cyanide administered by Hilary. That is 3375/6375= 0.53. 53%.

So, most likely, he died from the cyanide.

 

[Romsek]

So, most likely, he died from the cyanide.

I think he died from an overdose of hamberders and covfefe...