Do This by Logic

Again I am trying to get you to think, not blindly use formulas. You reach into a bowl that you know contains three fair coins and one double headed coin. You randomly pick one. Without looking at it, you flip it two times and get two heads. What's the chances you picked the bad coin?

...I thought it said 2 fair coins, not 3. So my answer would be 25%.


Ryan

I dont think 2 flips of heads, would be a big enough sample to change any percentage. There are 3 regular coins and one bad (1 out of 4) thats why I say 25%. Now if it came up 12 times in a row, then your odds would be almost 100%.

no sample size is sufficient enough to change the effective odds of which coin you pulled out. you could flip it a zillion times, get heads every time, and you could not be mathematically certain that you had pulled the double-headed coin. likely, but not certain.

The funny coin will be picked 1/4 of the time. A fair coin will be chosen 3/4 of the time and come up heads twice 1/4 of the time, which means this sequence occurs 3/4 * 1/4 = 3/16 of the time. Thus, HH will appear 3/16 + 1/4 = 7/16. So the probability that HH is caused by the funny coin is 1/4 / 7/16 = 4/7.

there are four apples in a basket


one of them is laced with sleeping potion


a guy picks one at midnight and eats it


he falls asleep


another guy takes a bite and he falls asleep


what is the probability that this is the funny apple?

1/4 assuming they would fall asleep at midnight regardless.

... is 4/7 !?

First try:


The chances you would initially pick a fair coin are 3/4. The chances that a fair coin would flip two heads in a row is 1/4. So the chances that you picked a fair coin AND flipped 2 heads in a row with it is 3/4 * 1/4 = 3/16. Therefore the chance that you picked the double-headed coin is 13/16, since 3/16 plus 13/16 = 1.0


Intuitively this seems high, so I am going to think about it a bit more while I go out for a bit of exercise.