You’re stranded in a rainforest, and you’ve eaten a poisonous mushroom. To save your life, you need an antidote excreted by a certain species of frog. Unfortunately, only the female frog produces the antidote. The male and female look identical, but the male frog has a distinctive croak. Derek Abbott shows how to use conditional probability to make sure you lick the right frog and get out alive.
If you chose to go to the clearing, you're right, but the hard part is correctly calculating your odds.
There are two common incorrect ways of solving this problem. Wrong answer number one: Assuming there's a roughly equal number of males and females, the probability of any one frog being either sex is one in two, which is 0.5, or 50%. And since all frogs are independent of each other, the chance of any one of them being female should still be 50% each time you choose. This logic actually is correct for the tree stump, but not for the clearing.
Wrong answer two: First, you saw two frogs in the clearing. Now you've learned that at least one of them is male, but what are the chances that both are? If the probability of each individual frog being male is 0.5, then multiplying the two together will give you 0.25, which is one in four, or 25%. So, you have a 75% chance of getting at least one female and receiving the antidote.
So here's the right answer. Going for the clearing gives you a two in three chance of survival, or about 67%. If you're wondering how this could possibly be right, it's because of something called conditional probability. Let's see how it unfolds. When we first see the two frogs, there are several possible combinations of male and female. If we write out the full list, we have what mathematicians call the sample space, and as we can see, out of the four possible combinations, only one has two males. So why was the answer of 75% wrong? Because the croak gives us additional information. As soon as we know that one of the frogs is male, that tells us there can't be a pair of females, which means we can eliminate that possibility from the sample space, leaving us with three possible combinations. Of them, one still has two males, giving us our two in three, or 67% chance of getting a female. This is how conditional probability works. You start off with a large sample space that includes every possibility. But every additional piece of information allows you to eliminate possibilities, shrinking the sample space and increasing the probability of getting a particular combination. The point is that information affects probability. And conditional probability isn't just the stuff of abstract mathematical games. It pops up in the real world, as well.
Computers and other devices use conditional probability to detect likely errors in the strings of 1's and 0's that all our data consists of. And in many of our own life decisions, we use information gained from past experience and our surroundings to narrow down our choices to the best options so that maybe next time, we can avoid eating that poisonous mushroom in the first place.
Một nhà thám tử đi cắt tóc. Trong thị trấn đó chỉ có 2 tiệm cắt tóc. Khi nhìn vào tiệm thứ nhất, nhà thám tử thấy một tiệm khá tồi tàn, đồ đạc vứt lung tung, đầu tóc của thợ cắt tóc thì là một kiểu tóc rất lạc hậu. Trong khi đó tiệm thứ hai rất gọn gàng, mọi thứ sạch sẽ, chủ tiệm cũng có đầu tóc cực kỳ hợp mốt. Nhà thám tử sau một hồi suy nghĩ thì quyết định đi vào tiệm thứ nhất để cắt tóc. Theo bạn lý do là vì sao?
Chỉ có 2 tiệm cắt tóc trong thị trấn nên suy ra đầu tóc của ông chủ tiệm thứ hai chính là do người thợ ở tiệm thứ nhất cắt (vì ông ta không thể cắt tóc cho chính mình được). Vì vậy người thợ đầu tiên là người có tay nghề cao hơn và cắt đẹp hơn.