Let the sample space be s 1 2 3 4 5 6 suppose the outcomes are equally likely


Introduction

Let the sample space be S = {1, 2, 3, 4, 5, 6} and suppose the outcomes are equally likely. Then, the probability of each outcome is 1/6.

What is a Sample Space?

The term sample space refers to the set of all outcomes of a random experiment. The sample space is usually denoted by S and the outcomes are usually denoted by s.

For example, if we roll a die, the sample space would be S={1,2,3,4,5,6} and an outcome would be s=4.

If we flip a coin, the sample space would be S={H,T} and an outcome would be s=H.

Note that the sample space is not the same as the set of all possible outcomes. The set of all possible outcomes is usually referred to as the event space and is denoted by E. For example, if we roll a die and flip a coin, the event space would be E={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6

What is an Event?


An event is a set of outcomes of a random experiment with a given probability. The sample space S of a random experiment is the set of all possible outcomes of that experiment.

For example, suppose we have a coin and we toss it three times. The sample space S for this experiment is {HHH, HHT, HTH, HTT, THH, THT, TTH,TTT}, where H represents “heads” and T represents “tails”.

The set {HHH} is an event since it is one of the outcomes in the sample space S. Another example is the set {HTT}. The empty set ∅ and S itself are also events since they are subsets of S.

What is the Probability of an Event?

The probability of an event is the chance that the event will happen. It is a number between 0 and 1.

An event that is certain to happen has a probability of 1.
An event that cannot happen has a probability of 0.

For example, the probability of rolling a 6 on a die is 1/6.

The probability of an event happening is usually written as a decimal or percentages. For example, the probability of it raining tomorrow could be written as 0.6 (60%) or 3/5 (three-fifths).

The sample space is the set of all possible outcomes of a random experiment. For example, if you roll a dice, the sample space would be {1,2,3,4,5,6}.

Theorem


If the sample space S is finite and the outcomes are equally likely, then the probability of any event E is given by:

P(E) = |E|/|S|

where |E| is the number of outcomes in E and |S| is the total number of outcomes in S.

Proof


We will use the fact that if two events A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

P(1 or 2 or 3 or 4 or 5 or 6) = P(1) + P(2) + P(3) + P(4)+ P(5)+ P(6)

P(s) = 1/6 * (1 + 1 + 1+ 1+ 1+ 1) = 1

Conclusion

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