- One way of thinking about probability is considering multiple possible worlds
- The world where A is the case, and the world where not-A is the case
- The base rate * likelihood of observation in that world = chance of being in that world
- The numbers you use are relative, not absolute (i.e. 0.2 and 0.8 ≡ 20 and 80 ≡ 1/5 and 4/5)
- Try to model the real world
- As finegrained as possible
1% of women in a certain group have breast cancer. 80% of women with breast cancer will get a positive mammogram result, and 9.6% of women without breast cancer will also get a positive result. If a woman in that group has a positive mammogram result, what are the chances she has breast cancer?
| Cancer | Not Cancer | |
|---|---|---|
| Base rate | 1 | 99 |
| LOO | 80 | 9.6 |
| Relative chance | 80 | 950.4 |
Chance of breast cancer = 80/(80+950.4) = 0.077
The chance of a biased coin is 50%. Of biased coins, 2/3 are biased towards heads (producing heads 3/4 of the time) and 1/3 are biased towards tails (producing tails 3/4 of the time). You flip a coin 3 times and get THT. What is the probability of the coin being fair?
| Fair | Hbias | Tbias | |
|---|---|---|---|
| Base rate | 3 | 2 | 1 |
| LOO (of getting THT) | 1/8 | 3/64 | 6/64 |
| Relative chance | 3/8 | 6/64 | 6/64 |
Chance of a fair coin = (1/8)/(1/8+3/64+6/64) = 8/17
You have a dresser with 8 drawers. You think the probability of your socks being in your dresser is 4/5. You have checked 6 drawers, and haven't found your socks. What are the chances your socks are in the next drawer you check?
| in d1 | d2 | d3 | d4 | d5 | d6 | d7 | d8 | outside of dresser | |
|---|---|---|---|---|---|---|---|---|---|
| Base rate | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| LOO (not in drawer 1) | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| LOO (not in drawer 2) | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| LOO (not in drawer 3) | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| LOO (not in drawer 4) | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| LOO (not in drawer 5) | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| LOO (not in drawer 6) | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
| Relative chance | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 |
Chance of socks in next drawer = 1/(1+1+2) = 1/4