Random Experiments

Random Experiments

Before rolling a die you do not know the result. This is an example of a random experiment. In particular, a random experiment is a process by which we observe something uncertain. After the experiment, the result of the random experiment is known. An outcome is a result of a random experiment. The set of all possible outcomes is called the sample space. Thus in the context of a random experiment, the sample space is our universal set. Here are some examples of random experiments and their sample spaces:

• Random experiment: toss a coin; sample space: S={heads,tails}$S=\{heads,tails\}$ or as we usually write it, {H,T}$\{H,T\}$.
• Random experiment: roll a die; sample space: S={1,2,3,4,5,6}$S=\{1,2,3,4,5,6\}$.
• Random experiment: observe the number of iPhones sold by an Apple store in Boston in 2015$2015$; sample space: S={0,1,2,3,⋯}$S=\{0,1,2,3,\cdots \}$.
• Random experiment: observe the number of goals in a soccer match; sample space: S={0,1,2,3,⋯}$S=\{0,1,2,3,\cdots \}$.

When we repeat a random experiment several times, we call each one of them a trial. Thus, a trial is a particular performance of a random experiment. In the example of tossing a coin, each trial will result in either heads or tails. Note that the sample space is defined based on how you define your random experiment. For example,

Example 
We toss a coin three times and observe the sequence of heads/tails. The sample space here may be defined as

S={(H,H,H),(H,H,T),(H,T,H),(T,H,H),(H,T,T),(T,H,T),(T,T,H),(T,T,T)}.$$S=\{(H,H,H),(H,H,T),(H,T,H),(T,H,H),(H,T,T),(T,H,T),(T,T,H),(T,T,T)\}.$$

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Our goal is to assign probability to certain events. For example, suppose that we would like to know the probability that the outcome of rolling a fair die is an even number. In this case, our event is the set E={2,4,6}$E=\{2,4,6\}$. If the result of our random experiment belongs to the set E$E$, we say that the event E$E$ has occurred. Thus an event is a collection of possible outcomes. In other words, an event is a subset of the sample space to which we assign a probability. Although we have not yet discussed how to find the probability of an event, you might be able to guess that the probability of {2,4,6}$\{2,4,6\}$ is50$50$ percent which is the same as 12$\frac{1}{2}$ in the probability theory convention.

Outcome: A result of a random experiment.
Sample Space: The set of all possible outcomes.
Event: A subset of the sample space.

Union and Intersection: If A$A$ and B$B$ are events, then A∪B$A\cup B$ and A∩B$A\cap B$ are also events. By remembering the definition of union and intersection, we observe that A∪B$A\cup B$ occurs if A$A$ or B$B$ occur. Similarly, A∩B$A\cap B$ occurs if both A$A$ and B$B$ occur. Similarly, if A1,A2,⋯,An$A_1,A_2,\cdots ,A_n$ are events, then the event A1∪A2∪A3⋯∪An$A_1\cup A_2\cup A_3\cdots \cup A_n$ occurs if at least one ofA1,A2,⋯,An$A_1,A_2,\cdots ,A_n$ occurs. The event A1∩A2∩A3⋯∩An$A_1\cap A_2\cap A_3\cdots \cap A_n$ occurs if all of A1,A2,⋯,An$A_1,A_2,\cdots ,A_n$ occur. It can be helpful to remember that the key words "or" and "at least" correspond to unions and the key words "and" and "all of" correspond to intersections.