Probability growth with time

[elite5chris]

I've been wondering about a simple question that I can't just google and get the answer to. Usually when we calculate probability, we know the number of possible outcomes. Say we toss a coin, there are 2 possible outcomes with one being head and one being tail. So the chance of getting a head is 1/2, WHEN you toss the coin. So maybe "tossing the coin" is a defined event. What if the event is continuous in time? What if say, instead of tossing the coin, I balance the coin so it rolls down the hills, and I ask the question, what is the chance of the coin losing its balance after rolling down the hills for 10 seconds, for 20 seconds, and so on? A more practical example would be what is the chance of getting into a car accident after driving for x amount of time? I would expect the theory that corresponds to this to be a growth of probability with respect to time where it reaches 100% when t->infinity.

 

[micromass]

I'm not sure what your question actually is. But events like the one you posted in the OP can be handled with continuous distributions (and even with discrete distributions). For example, check the exponential distribution: http://en.wikipedia.org/wiki/Exponential_distribution

 

[Stephen Tashi]

Taking the viewpoint of conditional probability, the usual way to look at things is that "The probability of event A given event B" does not change with time. If you want time to enter the picture you define a function that maps time to events. So "The probability of A given B(t)" can change with time because as time t changes, the event B(t) becomes a different event.

 

[Hornbein]

I've been wondering about a simple question that I can't just google and get the answer to. Usually when we calculate probability, we know the number of possible outcomes. Say we toss a coin, there are 2 possible outcomes with one being head and one being tail. So the chance of getting a head is 1/2, WHEN you toss the coin. So maybe "tossing the coin" is a defined event. What if the event is continuous in time? What if say, instead of tossing the coin, I balance the coin so it rolls down the hills, and I ask the question, what is the chance of the coin losing its balance after rolling down the hills for 10 seconds, for 20 seconds, and so on? A more practical example would be what is the chance of getting into a car accident after driving for x amount of time? I would expect the theory that corresponds to this to be a growth of probability with respect to time where it reaches 100% when t->infinity.

The Poisson distribution is what you are looking for.

 

[blue_raver22]

I can assure you that your question is not a simple one and there is no one-size-fits-all answer. The concept of probability growth with time is known as stochastic processes and it is a widely studied topic in mathematics and statistics. Stochastic processes deal with the random evolution of a system over time, where the outcome at any given time is uncertain.

In the case of tossing a coin, as you mentioned, there are only two possible outcomes and the probability of getting a head or tail remains constant at 1/2. However, when we introduce continuous time into the equation, the probability of an event occurring can change over time. In your example of balancing a coin on a hill, the probability of the coin losing its balance may increase as time goes on due to external factors such as wind or changes in the slope of the hill.

In practical situations, such as the chance of getting into a car accident after driving for a certain amount of time, the probability may indeed increase over time. This is because the longer you are driving, the more opportunities there are for potential accidents to occur. However, this does not mean that the probability will reach 100% at infinity. It is important to note that probability is a measure of likelihood and not a certainty. There is always a chance, no matter how small, that an event may not occur.

Stochastic processes are complex and involve advanced mathematical concepts, so it is not easy to provide a definitive answer without more specific details about the system in question. However, I hope this response has given you some insight into the concept of probability growth with time and the importance of understanding stochastic processes in scientific research.