Probability of eating one egg with salmonella

[Thecla]

ln Will Kurt's Kurt's book "Bayesian Statistics The Fun Way" he gives a problem at the end of a chapter
" Raw eggs have a 1/20,000 probability of having salmonella. If you eat two raw eggs what is the probability that you ate a raw egg with salmonella."

The online answer he gives:
"For this question you need the sum rule because if either egg has salmonella you will get sick.
prob(egg1)+prob(egg2)-prob(egg1) x prob(egg2) or
1/20000+1/20000-(1/20000)*(1/20000)=39,999/400,000,000 or a hair under 1/10000
I think I understand this.( you subtract probability of both eggs having salmonella)
But when I calculate getting ONE six throwing a pair of dice using this method I get
1/6+1/6-(1/6*1/6)=11/36.
This is the wrong answer.
The answer should be 10/36. Somebody told me you have to subtract 1/36 twice.If so why do you do it only once with eggs

 

[anuttarasammyak]

This is the wrong answer.

That seems the right answer. The probability to get at least one ONE by throwing a pair of dice is 11/36.
The cases are made of 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 3-1, 4-1, 5-1, 6-1 that counts 11.

 

[Hornbein]

ln Will Kurt's Kurt's book "Bayesian Statistics The Fun Way" he gives a problem at the end of a chapter
" Raw eggs have a 1/20,000 probability of having salmonella. If you eat two raw eggs what is the probability that you ate a raw egg with salmonella."

The online answer he gives:
"For this question you need the sum rule because if either egg has salmonella you will get sick.
prob(egg1)+prob(egg2)-prob(egg1) x prob(egg2) or
1/20000+1/20000-(1/20000)*(1/20000)=39,999/400,000,000 or a hair under 1/10000
I think I understand this.( you subtract probability of both eggs having salmonella)
But when I calculate getting ONE six throwing a pair of dice using this method I get
1/6+1/6-(1/6*1/6)=11/36.
This is the wrong answer.
The answer should be 10/36. Somebody told me you have to subtract 1/36 twice.If so why do you do it only once with eggs

You are calculating one or more eggs having salmonella, but exactly one six. So the cases are different. If you want one or more sixes, then 11/36 is correct.

If you want exactly one six, then it is 2*(six*not six) = 2*(1/6*5/6) = 10/36.

 

[FactChecker]

1/6 is the probability of getting a six, including cases where others have a six.
1/6+1/6 is the probability of a six in two die, but it double counts the case of getting two sixes.
1/6+1/6-1/36 removes the double counting and is the probability of getting one or two sixes.
1/6+1/6-2/36 removes any possibility of getting two sixes. It is the probability of getting exactly one six.

 

[hutchphd]

But what if Monty Hall offers to replace your egg with egg number three?

 

[Thecla]

I like your answer FACTCHECKER. But why isn't the probability of eating only one egg with salmonella
1/20000 +1/20000-(1/20000)(1/20000)-(1/20000)(1/20000). That is subtracting the squared term twice.

Regarding AMUTARRAsSAMYAK I am asking for only one six not at least one six. For only one six the answer is 10/36.
Part of the problem is how do you interpret "ate a raw egg with salmonella". I thought it meant only one egg with salmonella.

 

[DaveC426913]

Part of the problem is how do you interpret "ate a raw egg with salmonella". I thought it meant only one egg with salmonella.

Context! You cannot interpret a portion of a sentence and expect to get it right. The full sentence is "If you eat two raw eggs what is the probability that you ate a raw egg with salmonella."


"I ate a raw egg with salmonella" still resolves to TRUE even if both eggs were affected.

 

[FactChecker]

Part of the problem is how do you interpret "ate a raw egg with salmonella". I thought it meant only one egg with salmonella.

There are no problem-statement police, but the current problem statement is a little ambiguous. I would normally assume that "ate a raw egg with salmonella" really means one or more. If I wanted to say exactly one, I would say "exactly one raw egg with salmonella". If I wanted to include the possibility of two bad eggs, I would say "one or more eggs with salmonella",

 

[anuttarasammyak]

If one infected egg corresponds to one click of button to anticipated death and the next click cancels the first one, view of OP would make sense.

 

[FactChecker]

If one infected egg corresponds to one click of button to anticipated death and the next click cancels the first one, view of OP would make sense.

Ha! That's right. But in math problems, realism often doesn't apply. After all, math is the only place where Suzy can buy 42 watermelons and nobody asks why. ;-)

I think that the wording can also be clearer. I would interpret "an egg" as one or more because it doesn't specify "one egg". I would interpret "one egg" as exactly one egg. But better would be "one or more eggs" or "exactly one egg".

 

[pbuk]

Part of the problem is how do you interpret "ate a raw egg with salmonella". I thought it meant only one egg with salmonella.

That interpretation does not make sense:

because if either egg has salmonella you will get sick

and if both eggs have salmonella you will still get sick.

There could perhaps be some ambiguity if the question said "ate one raw egg with salmonella", but it doesn't.

 

[Thecla]

I think everything is clear now. Thanks. I did count the dice problem of getting one six(only 36 possible outcomes). I certainly did not want to make a a table with 20000 eggs and start counting those possibilities.