Chapter 7Probability7.1 Experiments, Sample Spaces and EventsStart with some definitions we will need in our study of probability.An EXPERIMENT is an activity with an observable result. Tossing coins, rollingdice and choosing cards are all probability experiments.The result of the experiment is called the OUTCOME or SAMPLE POINT. So the two possible outcomes from tossing a coin are H(heads) and T (tails).The set of all outcomes or sample points is called the SAMPLE SPACE of the experiment.An EVENT is a subset of a sample space. That is, an event can contain one or more outcomes that are in the sample space.Consider tossing a coin. The sample space is S = {H, T}. The events that are possible in this experiment are ∅, {H}, {T },S. So, whilethere are 2 outcomes in the sample space, there are 4 different events.If a 6-sided die is rolled, the sample space is S = {1, 2, 3, 4, 5, 6}.Sometimes we use a tree diagram to find all the possible outcomes of an experiment. Consider tossing a coin 3 times and noting theresult of each toss.HTHTHTHTHTHTHTHHTHTHHTTTHHTHTTTHTTTfirsttosssecond tossthirdtossoutcomeHHHSo we find there are 8 elements in the sample space. We would show this asS = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}0cJanice Epstein 1998, 1999, 2000. These notes may not be distributed for profit7.2 Definition of Probability 2Find the event E where E = {x|x has exactly one head} E = {HTT,THT,TTH}Find the event E where E = {x|x has two or more heads} E = {HHT, HT H, THH, HHH}Find the event E where E = {x|x has more than 3 heads} E = ∅Another common sample space you must be familiar with is that from rolling two dice. We put this in a chart. You can think of the diceas being two different colors to avoid missing any outcomes.1-1 2-1 3-1 4-1 5-1 6-11-2 2-2 3-2 4-2 5-2 6-21-3 2-3 3-3 4-3 5-3 6-31-4 2-4 3-4 4-4 5-4 6-41-5 2-5 3-5 4-5 5-5 6-51-6 2-6 3-6 4-6 5-6 6-6These sample spaces have all been finite. That is, we can list all the elements. An infinite sample space has to be described, you can’tlist all the elements:What is the sample space for the time spent working on a homework set?S = {t|t ≥ 0, t in minutes}Describe the event of spending between one and two hours on a homework set.E = {t|60 <t<120}7.2 Definition of ProbabilityWhen we toss a fair coin the two outcomes in the sample space S = {H, T } are equally likely, so the probability of each outcome is1/2. We would write this as P ({H})=1/2and P ({T })=1/2. This is a THEORETICAL PROBABILITY based on the sample spacehaving equally likely outcomes. In general, this is the way we will find probability, by using a sample space of EQUALLY LIKELYOUTCOMES. The probability of an event, P (E) is a number between 0 and 1, 0 ≤ P (E) ≤ 1. An event with a probability of 0 isIMPOSSIBLE and an event with a probability of 1 is CERTAIN. The closer P (E) is to 1, the more likely the event is to happen.We can also calculate the EMPIRICAL PROBABILITY of an event by doing an experiment many times. For example, you could toss acoin and note how many times it comes up heads (shown in book) or you could roll a die and count how many times a 1 is rolled.number of tosses (m)number of 1’s rolled (n) relative frequency (n/m)20 3 3/20 = .1510018 18/100 = .181000160 160/1000 = .16100001662 1662/10000 = .1662If the die is fair, the theoretical probability is 1/6 = .16666666..... We would expect that if we did the experiment enough times we wouldapproach the theoretical value rather closely if the die were fair. In fact, the empirical probability that an event occurs is the relativefrequency that the event occurs as the number of experiments get very large (n →∞). So, if the die (or coin) was not fair, we couldstill find the probability of each outcome by doing lots of experiments. The balls used in the Lotto game are checked this way - they domany many draws and make sure each ball has a 1/50 chance of being drawn.A third way of “calculating” probability is called SUBJECTIVE PROBABILITY. That is when an expert estimates the probability ofsomething happening based on their opinion. Betting lines, economic predictions and weather forcasting are all based on this kind ofprobability.When finding probability theoretically we will always start by finding the sample space. You must be certain that it is a UNIFORMSAMPLE SPACE - each of the outcomes are equally likely. We write S = {s1,s2, ..., sn}, a sample space with n equally likely outcome.The events {s1}, { s2} ... {sn} are called SIMPLE events because they consist of exactly one outcome. Notice these simple events are7.2 Definition of Probability 3MUTUALLY EXCLUSIVE as only one can occur. The probability of each simple event occuring is the same,P (s1)=P(s2)=...P (sn)=1/nWe can put the probability of each event in a table called a PROBABILITY DISTRIBUTION TABLE:outcomeprobability{s1} P (s1)=1/n = P1{s2} P (s2)=1/n = P2...{sn} P (sn)=1/n = PnNotice the following properties of probabilitydistributions:1. 0 ≤ P (si) ≤ 12. P1+ P2+ ...Pn=13. P ({si}∪{sj})=Pi+Pj,i6= jSo we can go back to our sample spaces from section 7.1 and write the probability distribution table:Toss a coin three times. There are 8 equally likely outcomes:simple eventProbability{HHH} 1/8{HHT }1/8{HT H}1/8{HT T}1/8{THH}1/8{THT}1/8{TTH}1/8{TTT}1/8Notice each event is mutually exclusive. this must be true of the list in your probability distribution table or it won’t work.We can also use our relative frequency interpretation of empirical probabilities for doing an experiment and assigning probability.Suppose the instructor of a class polled the students about the number of hours spent per week studying math during the previous week.The results weretime studying (hours)number of students Probability (relative frequency)0 ≤ x ≤ 2 69 69/309 = .22332 <x≤4128 128/309 = .41424 <x≤668 68/309 = .22016 <x≤830 30/309 = .0971x>814 14/309 = .0453total 309 1Since the catagories are mutually exclusive, we can find the probability that a student studies more than 4 hours per week asP (x>4) = P (4 <x≤6) + P (6 <x≤8) + P (x>8)= .2201 + .0971 + .0453 = .3625What is the probability of rolling a sum 2 or a sum of 12 using two fair die?Remember the sample space has 36 outcomes. Look at the sample space and find that the sum of 2 happens once and the sum of 12happens once and they are mutually exclusive.P =1/36 + 1/36 = 2/36 = 1/18What is the probability of rolling a sum of 7? Look at the sample space:P =1/36 + 1/36 + 1/36 + 1/36 +
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