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Question: The probability that a randomly chosen record has an error is 0.05. An error in a record is either large or small. Two out of ten errors, on average, are large.

a. What is the probability that a randomly chosen record has a large error

Using Wolfram Mathematica, my sample space is

Sample Space

Since the probability of any record having an error is 0.05 with $\frac{2}{10}$ being a large error, it must be true that

$\frac{2}{10}$ $\cdot$ 0.05= 0.01 is a large error and the probability of a small error is 0.99.

b. Given that a randomly chosen record does not have a large error, what is the probability that it does not have an error.

b. is a conditional probability.

In my attempt, we have

$P\left ( Null Error | Small Error \right )=\frac{P\left ( Null Error \cap Small Error \right )}{P\left ( Small Error \right )} = \frac{0.99}{0.76}$

But this looks odd. It doesn't make sense for me to talk of small or large error in the null error case.

Any help is appreciated.

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  • $\begingroup$ Numbers in "Null Error" column are larger than 1, therefore not probabilities. How did you create that table? $\endgroup$ – LmnICE Jan 12 '18 at 10:40
  • $\begingroup$ @LmnICE It was a computational error. I have since made the correction. $\endgroup$ – Physkid Jan 12 '18 at 10:44
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Since an error is either large or small (and cannot be both at the same time, meaning the events have the null set as intersection), you have $$\mathbb{P}[\text{error}]=\mathbb{P}[\text{small error}] + \mathbb{P}[\text{small error}].$$

As such, $\mathbb{P}[\text{small error}] = 0.04$, and $\mathbb{P}[\text{no error}] = 1 - \mathbb{P}[\text{error}] = 0.99$

Regarding b., not having a large error is not the same as having a small error: $$\{\text{no large error}\} = \{\text{small error}\} \cup \{\text{no error}\},$$ therefore $\mathbb{P}[\text{no large error}] = 0.04 + (1-0.05) = 0.99$, which means $$ \mathbb{P}[\text{no error}\mid\text{no large error}] =\dfrac{\mathbb{P}[\text{no error}]}{\mathbb{P}[\text{no large error}]} = \dfrac{0.95}{0.99} $$

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  • $\begingroup$ A burning question that rest in me is 'Regarding b., not having a large error is not the same as having a small error' considering that the axiom indicates that an error is either large or small. $\endgroup$ – Physkid Jan 12 '18 at 11:10
  • $\begingroup$ @Physkid there are three possible states: no error, small error and large error. Problem states that it’s not a large error, so it’s either a small error or no error. That’s what Emil meant. $\endgroup$ – LmnICE Jan 12 '18 at 11:16
  • $\begingroup$ The axiom indicates that an error is either large or small given that an error happens. Since your probability of an error happening is 0.05 and not 1, there is a rather good probability (0.95) that no error happens at all. $\endgroup$ – Emil Jan 12 '18 at 11:16
  • $\begingroup$ @LmnICE Thank you guys. This is now clear. I should have been more careful. $\endgroup$ – Physkid Jan 12 '18 at 11:17

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