3 ms·

The highly recommended classic: http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes

deskamess 16y ago
I could not solve the following problem from the article. Suppose that 40% of the eggs are painted blue, 5/13 of the eggs containing pearls are painted blue, and 20% of the eggs are both empty and painted red. What is the probability that an egg painted blue contains a pearl? Any hints/ideas?
- tel 16y ago
  It requires a little bit of indirection. You'll find that you could easily solve the equation if you only knew one more quantity. So call the unknown "X" and keep going. You'll be able to form an equation that constrains X to a single value. FWIW, the answer I get has three non-zero decimal digits.
- defen 16y ago
  Pr(pearl|blue) = Pr(blue|pearl) * Pr(pearl) / Pr(blue) Two are given outright, and you should be able to compute the third based on the available data.
- jules 16y ago
  Make a table: | blue | red | --------|------|------|----- pearls | | | --------|------|------|----- empty | | | --------|------|------|----- | | | 100 We know that 40% of the eggs are blue, so we fill in 40 at the bottom row in the blue column. We now also know that 60% are red. Similarly, 20% of eggs are empty & red, so we fill in 20 there. Now we can calculate that 60% - 20% = 40% are pearls & red: | blue | red | --------|------|------|----- pearls | | 40 | --------|------|------|----- empty | | 20 | --------|------|------|----- | 40 | 60 | 100 Now comes the final bit of data: 5/13 of the eggs containing pearls are painted blue: | blue | red | --------|------|------|----- pearls | X | 40 | 40+X --------|------|------|----- empty | | 20 | --------|------|------|----- | 40 | 60 | 100 With X = 5/13*(40+X), or X = 100/3. | blue | red | --------|-------|------|----- pearls | 100/3 | 40 | 220/3 --------|-------|------|----- empty | 20/3 | 20 | 80/3 --------|-------|------|----- | 40 | 60 | 100 Now we can answer the question: what is the probability that an egg painted blue contains a pearl? For every 40 blue eggs, 100/3 contain pearls. So the answer is 100/3/40 = 5/6.
  - defen 16y ago
    I think you made an arithmetic mistake here. > With X = 5/13 (40+X), or X = 100/3. Should be: With X = 5/13 (40+X), or X = 25
    - jules 16y ago
      You are right. I am unable to edit that post unfortunately. The final table becomes: | blue | red | --------|------|------|----- pearls | 25 | 40 | 65 --------|------|------|----- empty | 15 | 20 | 35 --------|------|------|----- | 40 | 60 | 100 And the final answer is 5/8.
- deskamess 16y ago
  Thanks for the responses (3 at this time). I will try and work through them.

I could not solve the following problem from the article. Suppose that 40% of the eggs are painted blue, 5/13 of the eggs containing pearls are painted blue, and 20% of the eggs are both empty and painted red. What is the probability that an egg painted blue contains a pearl? Any hints/ideas?

tel 16y ago
It requires a little bit of indirection. You'll find that you could easily solve the equation if you only knew one more quantity. So call the unknown "X" and keep going. You'll be able to form an equation that constrains X to a single value. FWIW, the answer I get has three non-zero decimal digits.
defen 16y ago
Pr(pearl|blue) = Pr(blue|pearl) * Pr(pearl) / Pr(blue) Two are given outright, and you should be able to compute the third based on the available data.
jules 16y ago
Make a table: | blue | red | --------|------|------|----- pearls | | | --------|------|------|----- empty | | | --------|------|------|----- | | | 100 We know that 40% of the eggs are blue, so we fill in 40 at the bottom row in the blue column. We now also know that 60% are red. Similarly, 20% of eggs are empty & red, so we fill in 20 there. Now we can calculate that 60% - 20% = 40% are pearls & red: | blue | red | --------|------|------|----- pearls | | 40 | --------|------|------|----- empty | | 20 | --------|------|------|----- | 40 | 60 | 100 Now comes the final bit of data: 5/13 of the eggs containing pearls are painted blue: | blue | red | --------|------|------|----- pearls | X | 40 | 40+X --------|------|------|----- empty | | 20 | --------|------|------|----- | 40 | 60 | 100 With X = 5/13*(40+X), or X = 100/3. | blue | red | --------|-------|------|----- pearls | 100/3 | 40 | 220/3 --------|-------|------|----- empty | 20/3 | 20 | 80/3 --------|-------|------|----- | 40 | 60 | 100 Now we can answer the question: what is the probability that an egg painted blue contains a pearl? For every 40 blue eggs, 100/3 contain pearls. So the answer is 100/3/40 = 5/6.
- defen 16y ago
  I think you made an arithmetic mistake here. > With X = 5/13 (40+X), or X = 100/3. Should be: With X = 5/13 (40+X), or X = 25
  - jules 16y ago
    You are right. I am unable to edit that post unfortunately. The final table becomes: | blue | red | --------|------|------|----- pearls | 25 | 40 | 65 --------|------|------|----- empty | 15 | 20 | 35 --------|------|------|----- | 40 | 60 | 100 And the final answer is 5/8.
deskamess 16y ago
Thanks for the responses (3 at this time). I will try and work through them.