# Probability - Binominal distribution

MarcMTL

## Homework Statement

A factory makes glass bottles, and 15.0% of the bottles are considered defective. A defective bottle will break 60.0% of the time when dropped from a controlled height, whereas a non-defective bottle will only break 10.0% of the time in the same conditions.

You pick a single bottle at random and drop it four times in a row. It doesn't break. What is the probability that it is a defective bottle?

## The Attempt at a Solution

If a defective bottle breaks 60% of the time, I calculated the probability of it surviving four consecutive drops: (1-0.60)4 = 0.0256 (2.56%)

This is where I get doubtful, I'd tend to say that if 20% of the bottles are defective, and we picked a single bottle at random, that the probability that the bottle is defective would be:

0.0256 * 0.20 = 0.00512 (0.512%)

Not the right answer. Also, I feel that the last step was incorrect.

Any help would be greatly appreaciated!

Marc

hgfalling
Try thinking about the whole population of bottles.

X% are not defective.
Y% are defective.

Now you pick a bottle at random and drop it four times.
What can happen?

It could be defective and survive. (a%)
It could be not-defective and survive (b%)
It could be defective and break (c%)
If could be not-defective and break (d%)

Now of course a+b+c+d = 100%. And we know that what actually happened was that our bottle didn't break, so we are actually only looking at (a+b)% of the population.

That should get you started..

Homework Helper
Another way is...

S: the bottle survived 4 drops
D: the bottle is defective

Find these probabilities:
P(S:D), P(S:!D), P(D), P(S).

Then use Bayes Theorem to find P(D:S).
The text passage contains a probability problem involving defective and non-defective glass bottles, and the probability of a bottle breaking when dropped. The problem is presented as a homework question, and the student provides an attempt at a solution, which is then critiqued by another user who suggests a different approach using Bayes' Theorem.
single_question_with_answer

the passage contains a single mathematical question, and while the student's attempt at a solution is not entirely correct, an answer is provided in the form of a suggested method to solve the problem using bayes' theorem.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Probability - Binominal distribution

MarcMTL

## Homework Statement

A factory makes glass bottles, and 15.0% of the bottles are considered defective. A defective bottle will break 60.0% of the time when dropped from a controlled height, whereas a non-defective bottle will only break 10.0% of the time in the same conditions.

You pick a single bottle at random and drop it four times in a row. It doesn't break. What is the probability that it is a defective bottle?

## The Attempt at a Solution

If a defective bottle breaks 60% of the time, I calculated the probability of it surviving four consecutive drops: (1-0.60)4 = 0.0256 (2.56%)

This is where I get doubtful, I'd tend to say that if 20% of the bottles are defective, and we picked a single bottle at random, that the probability that the bottle is defective would be:

0.0256 * 0.20 = 0.00512 (0.512%)

Not the right answer. Also, I feel that the last step was incorrect.

Any help would be greatly appreaciated!

Marc

hgfalling
Try thinking about the whole population of bottles.

X% are not defective.
Y% are defective.

Now you pick a bottle at random and drop it four times.
What can happen?

It could be defective and survive. (a%)
It could be not-defective and survive (b%)
It could be defective and break (c%)
If could be not-defective and break (d%)

Now of course a+b+c+d = 100%. And we know that what actually happened was that our bottle didn't break, so we are actually only looking at (a+b)% of the population.

That should get you started..

Homework Helper
Another way is...

S: the bottle survived 4 drops
D: the bottle is defective

Find these probabilities:
P(S:D), P(S:!D), P(D), P(S).

Then use Bayes Theorem to find P(D:S).
user
The text passage contains a probability problem involving defective and non-defective glass bottles, and the probability of a bottle breaking when dropped. The problem is presented as a homework question, and the student provides an attempt at a solution, which is then critiqued by another user who suggests a different approach using Bayes' Theorem.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question, and while the student's attempt at a solution is not entirely correct, an answer is provided in the form of a suggested method to solve the problem using Bayes' Theorem.