Probability: Random Model Behavior

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A probability model is a mathematical representation of a random phenomenon occurring. Probability is the study of long term behavior of random events in which outcomes cannot be predicted. During the experiment in the long run, patterns of outcomes emerge. A probability model includes its sample space, events that occur in the sample space, and possibilities associated with each event. When two events are independent there is no correlation of one event occurring for another event to occur. The first event does not influence the outcome of the second. This can be seen in the example of flipping a coin. When a coin is flipped and lands on heads three times determining which outcome will occur on the fourth flip would be an assumption. Just because the coin landed three times on heads does not mean that the fourth coin flip will land on heads. Actually, the chances of either event, (heads or tails), occurring is 50% or 0.5 or ½.
In the sample space of tossing two fair dice there we gather 36 different permutations. In sample space we can identify distinct events within the 36 permutations. One distinct event is having an outcome of two fair dices that have an outcome in which two dices sum are factors of 4. The outcomes that have factors of 4 are, (A, C), (B, B), (C, A), (D, D), (F, B), (B, F), (E, C), and (C, E). There are 8 summed permutations that equal 4. In the event of two fair dice being thrown would be an 8/36 chance, which is simplified as 2/9, which is 22% chance of receiving an outcome with the factor of 4. A second event is receiving an outcome of both dice receiving an outcome of both dices landing on a number less than 3. In the outcomes of this distinct event we have (A, A1), (A, B), (B, A), and (B, B1). We have a total of four outcomes in this distinct event. There is a 4/36, which is simplified as 1/9, or 11% chance of receiving an outcome where both independent dice are less than 3. Both
Probability is important for a stage in the study of treatments for many dread diseases. If a drug is being researched in its developmental stage, researchers calculate the maximum tolerated dose, whereas the dose has more toxic effects rather than health benefits.

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