TITLE: If the probability of a dog barking one or more times in a given hour is 84%, then what is the probability of a dog barking in 30 minutes?
QUESTION [4 upvotes]: Poorly worded title but I don't know what the nature of this probability question is called.
I was asked a question:
If the probability of a dog barking one or more times in a given hour is 84%, then what is the probability of a dog barking in 30 minutes?
Since I was told that the first solution I wanted to jump to of 42% is incorrect, I was then presented with the following steps:
The chance of the dog not barking in a given hour is 1-84% = 16%
If the chance of a dog not barking over the course of 2 units - 2 half hours for a total of one hour, then x * x = 16%.
Thus, the probability that the does does NOT bark in 30 minutes is $\sqrt{16%}$ = 40%. Therefore, the probability of the dog barking in a given 30 minutes is 1-40% = 60%.
Question1: Is this correct?
Question2: Rather than work with the inverse probability 16%, surely I can apply the same logic with just 84% and arrive at the same answer? So, if the probability of the dog barking one or more times in an hour is 84%, then this 84% could also be represented as the probability of two 30 minute instances of a dog barking at least once in each instance. In that case:
p(dog barks in 1st half hour AND dog barks in second half hour) = 84%.
Thus the chance of the dog barking in the first half hour is is $\sqrt{0.84}$ = 91.65%.
91.65% does not equal 60% which is what I arrived at by going the negative probability route. I was expecting both numbers to match.
What is the correct way to calculate the probability of a dog barking over 30 minutes if we know that the probability of the dog barking over an hour is 84%?

REPLY [2 votes]: The analysis leading to 60% is correct.  Your analysis does not take into account possibility of two barks in the same half hour, but none in the other.  
Also both attempts at analysis assumes barks are independent.  I doubt if most dogs behave that way.
{"set_name": "stack_exchange", "score": 4, "question_id": 3194064}