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Unformatted text preview: CHAPTER 2: Probability Sample Space: 2.1 A random experiment is an experiment in which the outcome varies in an unpredictable fashion when the experiment is repeated under the same conditions. Examples • Experiment E 1 : Toss a coin three times and note the sequence of heads and tails. • Experiment E 2 : Toss a coin three times and note the number of heads • Experiment E 3 : A block of information is transmitted repeatedly over a noisy channel until an error free block arrives at the receiver. Count the number of transmissions required. • Experiment E 4 : Measure the lifetime of a given computer memory chip in a specified environment. Since random experiments do not consistently yield the same result, it is necessary to determine what the set of possible results can be. The sample space S of a random experiment is defined as the set of all possible outcomes. Note: When we perform a random experiment, one and only one outcome occurs. The samples spaces corresponding to the experiments in the last Example are given below: • Experiment E 1 : Toss a coin three times and note the sequence of heads and tails. • Experiment E 2 : Toss a coin three times and note the number of heads • Experiment E 3 : A block of information is transmitted repeatedly over a noisy channel until an error free block arrives at the receiver. Count the number of transmissions required. • Experiment E 4 : Measure the lifetime of a given computer memory chip in a specified environment. Random experiments involving the same experimental procedure may have different sample spaces as shown by Experiments E 1 and E 2 . 1 A tree diagram: Example: The three balls numbered 1 to 3 in an urn are drawn at random one at a time until the urn is empty. The sequence of the ball numbers is noted. Find the sample space. Events: 2.2 We are usually not interested in the occurrence of specific outcomes, but rather on the occurrence of some event (i.e. whether the outcome satisfies certain conditions). Example: Experiment: Determine the value of a voltage waveform at time t 1 S = (∞ , ∞ ). We might be interested in the event “voltage is negative” which corresponds to (∞ , 0). The event occurs if and only if the outcome of the experiment is in this subset. For this reason we define an event as a subset of S Example: Experiment E 3 : A block of information is transmitted repeatedly over a noisy channel until an errorfree block arrives at the receiver. Count the number of transmissions required. S 3 = A = Fewer than 10 transmissions are required = Ex. 10 on p. 30: An engineering firm is hired to determine if certain waterways in Virginia are safe for fishing. Samples are taken from three rivers....
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 Spring '11
 rsharma

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