Probability of eating one egg with salmonella

In summary, the conversation discussed the probability of eating a raw egg with salmonella, with a given probability of 1/20,000 for each egg. The question asked was if someone ate two raw eggs, what is the probability of eating at least one egg with salmonella. The online answer provided a solution using the sum rule and the correct answer was found to be 1/10000. The conversation also discussed a similar problem involving rolling two dice and getting at least one six, with a solution of 11/36 using the sum rule. It was clarified that this solution counts for getting one or more sixes, and the correct solution for getting exactly one six is 10/36. There was some confusion regarding the interpretation
  • #1
Thecla
135
10
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
 
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  • #2
Thecla said:
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.
 
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  • #3
Thecla said:
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.
 
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  • #4
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.
 
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  • #5
But what if Monty Hall offers to replace your egg with egg number three?
 
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  • #6
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.
 
  • #7
Thecla said:
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.
 
  • #8
Thecla said:
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",
 
  • #9
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.
 
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  • #10
anuttarasammyak said:
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".
 
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  • #11
Thecla said:
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:
Thecla said:
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.
 
  • #12
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.
 

Related to Probability of eating one egg with salmonella

1. What is the probability of getting sick from eating one egg with salmonella?

The probability of getting sick from eating one egg with salmonella depends on various factors such as the amount of salmonella present in the egg, the individual's immune system, and the overall cleanliness and hygiene of the egg. However, on average, the probability is estimated to be around 1 in 20,000 eggs.

2. How does salmonella get into eggs?

Salmonella can get into eggs through various sources such as contaminated chicken feces, contaminated surfaces or equipment during egg production, or from the ovaries of infected hens. It is important to properly handle and cook eggs to prevent the risk of salmonella contamination.

3. Can you tell if an egg has salmonella just by looking at it?

No, you cannot tell if an egg has salmonella just by looking at it. Salmonella bacteria are invisible to the naked eye, so it is impossible to determine if an egg is contaminated just by its appearance. It is important to properly cook eggs to kill any potential bacteria.

4. How can I reduce the risk of getting sick from eating eggs with salmonella?

The best way to reduce the risk of getting sick from eating eggs with salmonella is to handle and cook them properly. This includes washing your hands before and after handling eggs, cooking eggs thoroughly until the yolk and whites are firm, and avoiding cross-contamination with other foods.

5. What are the symptoms of salmonella poisoning?

The symptoms of salmonella poisoning typically include diarrhea, abdominal cramps, fever, and vomiting. These symptoms can range from mild to severe and usually last for 4-7 days. In some cases, hospitalization may be required for more severe cases, especially in young children, older adults, and individuals with weakened immune systems.

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