When we observe the traffic lights, it often feels like the light is never green. But is this really true? What causes this phenomenon? These questions are related to the concept of randomness. Let's delve into the vocabulary associated with random phenomena.
Random Phenomenon and Sample Space
A random phenomenon is a situation in which different outcomes can possibly occur, but we don't know which particular outcome will happen. Each instance of observing a random phenomenon is called a trial. At each trial, the value of the random phenomenon's outcome is noted, and this is called the probability of the trial's outcome. The collection of all possible outcomes of a random phenomenon is called a sample space.
Sample Space Formula and Examples
The sample space, denoted as S, contains all the possible outcomes of a random phenomenon. For example, if you flip a coin once, the sample space is {H, T} because there are only two possible outcomes - heads or tails.
Probability
The probability of any outcome of a random phenomenon is a number between 0 and 1. The law of large numbers states that when we repeat a random process a large number of times, the proportion of times that an event occurs will tend to the probability of the event. This law assumes that the random phenomenon being studied doesn't change, and all the trials of the event are independent.
Coin Toss Example
For example, if we toss a coin 10 times and we get heads 5 times, the probability of getting heads on one toss is 5/10 or 0.5.
Modeling Probability
Probability can be modeled from a theoretical standpoint, not just from observation. Theoretical probability is based on the construction of a model and the application of mathematical concepts to determine the likelihood of certain outcomes.
Probability Model and its Rules
The theoretical probability of an event is the number of favorable outcomes divided by the total possible outcomes. It is represented as P(A) = # outcomes in event A / # possible outcomes. The probability of an event can also be calculated using the rule P(A) = 1 - P(A²) or the rule P(A U B) = P(A) + P(B) if A and B are disjoint events.
In conclusion, weather is often used as an example of a random phenomenon because its outcomes can vary, and we can never be entirely certain what the weather will be like on any given day. Randomness and probability are important concepts in statistics and are used to make predictions and analyze uncertainty in various real-world situations.
