B Calculation of probability

1. May 20, 2017

Mr Real

I recently read a question. It was: If 100 birds are sitting in a circle and all of them peck a bird either on their left or on their right randomly, what's the expected number of birds that will be unpecked? The answer to this is 25 birds as probability of not being pecked is 25%. The logic went like this: The probability of a bird not getting pecked from right side will be 50% and same with the left too and then we have to halve this to get the answer. But why did we have to halve to get the probability, I mean is there any formula or axiom for this that it's how probability is treated? I know it's a very basic question and we can all solve such problems without even thinking about it, but what is the reasoning behind this that when we considered 2 sides instead of 1 we had to divide the probability by 2?

2. May 20, 2017

Buzz Bloom

Hi Mr real:

The "halving" is a coincidence. There are two independent events: (1) what the left bird will do. (2) what the right bird will do. Each has a probability of 1/2 that they will not peck the bird in the middle. The probability that two independent events will occur is the product of their two probabilities.

Regards,
Buzz

3. May 20, 2017

Mr Real

Thank you very much. Just the answer I was looking for.