Probability, Laws of

Experiments, Outcome Sets, and Events

Probability laws are defined in the context of an experimentwhose outcome is due to chance. The set of all possible outcomes for a particular experiment is the outcome set,or sample space,and is commonly denoted by Ω. An eventis any subset of the outcome set Ω. Examples of experiments and their corresponding outcome sets are given below; each is accompanied by two event examples.

Experiment 1:Flip a coin and record which side, heads or tails, lands face up. In this case, the sample space consists of two elements: heads and tails. The outcome set could be denoted Ω = {H, T}. If E1represents the subset {T} of Ω, then E1is the event that a tail is flipped. If E2represents the subset {H} of Ω, then E2is the event that a head is flipped.

Experiment 2:Flip a coin twice and record the side that lands face up in the first flip, and then in the second flip. Here, there are four possible outcomes: Ω = {HH, HT, TH, TT}, where, for example, TH represents the outcome where a tail is observed on the first flip and a head on the second. If E1={HH, TT}, then E1denotes the event that the two flips are in agreement. If E2= {TH, TT}, then E2denotes the event that a tail was flipped first.

Experiment 3:Flip a coin until the first head appears. In this case, the number of outcomes is infinite, but countable: Ω = {H, TH, TTH, TTTH,…}. If E1is the event that at most two tails are observed before the first head, then E1is equal to the subset {H, TH, TTH} of Ω. If E2is the event that exactly three tails are observed, then E2={TTTH}.

Experiment 4:Roll a six-sided die so that it comes to rest on a table. Record the number of spots facing up. In this example, the outcome set is numeric: Ω = {1,2,3,4,5, 6}. If E1is the event that a number greater than 2 is rolled, then [Page 1105]E1={3, 4, 5, 6}. If EF2is the event that an even number is rolled, then E2= {2, 4, 6}.

Experiment 5:Roll a six-sided die so that it comes to rest on a round table. Record the distance from the die to the center of the table. If ris the radius of the table, then the experimental outcome must be a real number from 0 to r;hence, the outcome set is infinite and uncountable: Ω = [0,r). Notice that this interval implies that a die rolling a distance ror greater from the center of the table will fall off the table (and hence be ignored as it does not fit the definition of this experiment). If E1is the event that the die lands closer to the center of the table than to the nearest edge, then E1=[0, r/2). If E2is the event that the die lands in the outer half of the table area, then

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