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Probability

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Probability

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} or as we usually write it, $\{H, T\}$ .
Random experiment: roll a die; sample space: $S = \{1, 2, 3, 4, 5, 6\}$
Random experiment: observe the number of iPhones sold by an Apple store in Boston in 20152015; sample space: $S = \{0, 1, 2, 3 .. .\}$
Random experiment: observe the number of goals in a soccer match; sample space: $S = \{0, 1, 2, 3 .. .\}$

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.

Check Your Understanding!

Select the possible sample space when we toss a coin three times and observe the sequence of heads/tails.

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

How Did I Do?

Check Your Understanding

The sample space $S$ is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: $(1, 4)$ ).

1. Find the sample space, $S$ .

		$S = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)\}$
		$S = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)\}$
		$S = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3)\}$
		$S=\{$ $(0, 1), (0, 2), (0, 3), (0, 4), (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)$ $(2, 4), (3, 1), (3, 2), (3, 3), (3, 4)$ $}$
		$S=\{$ $(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4),$ $(3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)$ $}$

How Did I Do?

Probability

We assign a probability measure $P (A)$ to an event $A$ . This is a value between $0$ and $1$ that shows how likely the event is. If $P (A)$ is close to $0$ , it is very unlikely that the event $A$ occurs. On the other hand, if $P (A)$ is close to $1$ , $A$ is very likely to occur. The main subject of probability theory is to develop tools and techniques to calculate probabilities of different events. Probability theory is based on some axioms that act as the foundation for the theory, so let us state and explain these axioms.

Axioms Of Probability

Axiom 1: For any event $A$ , $P (A) \geq 0$ .
Axiom 2: Probability of the sample space $S$ is $P (S) = 1$ .
Axiom 3: If $A_{1}, A_{2}, A_{3}, .. .$ are disjoint events, then $P (A_{1} \cup A_{2} \cup A_{3} .. .) = P (A_{1}) + P (A_{2}) + P (A_{3}) + .. .$

Let us take a few moments and make sure we understand each axiom thoroughly.

The first axiom states that probability cannot be negative. The smallest value for $P (A)$ is zero and if $P (A) = 0$ then the event $A$ will never happen.
The second axiom states that the probability of the whole sample space is equal to one, i.e., $100$ percent. The reason for this is that the sample space $S$ contains all possible outcomes of our random experiment. Thus, the outcome of each trial always belongs to $S$ , i.e., the event $S$ always occurs and $P (S) = 1$
The third axiom is probably the most interesting one. The basic idea is that if some events are disjoint (i.e., there is no overlap between them), then the probability of their union must be the summations of their probabilities.

Check Your Understanding

In a presidential election, there are four candidates. Call them A, B, C, and D. Based on our polling analysis, we estimate that A has a $10$ percent chance of winning the election, while B has a $20$ percent chance of winning. What is the probability that A or B wins the election?

$P (A wins or B wins) =$

How Did I Do?Try Another

Number Help

You can use the math app below to test the outcomes for rolling two dice and view the probabilities of attaining those events.

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Probability

Probability

Random Experiments

Select the possible sample space when we toss a coin three times and observe the sequence of heads/tails.

The sample space S is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: 1,4).

Probability

The sample space $S$ is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: $(1, 4)$ ).