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17A-Empirical Probability and 17B-Theoretical Probability

Learn the basics of theoretical and empirical probability, including sample spaces, event occurrences, and calculating probabilities. Explore examples with dice rolls, card decks, and experimental probability using relative frequencies.

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17A-Empirical Probability and 17B-Theoretical Probability

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  1. 17A-Empirical Probability and 17B-Theoretical Probability Some newvocabulary…

  2. TheoreticalProbability • Sample space (U) = the list of all possibleoutcomes • Ex: rolladie. Sample space U = 1, 2, 3, 4, 5,6 • n(U) = the total number of items in sample spaceU • Ex: roll a die… n(U) =6 • n(A) = “success” = the number of times eventA • occurs in the samplespace • Ex: Event is rolling a 4 on a fair die (fair meanseach • outcome is equally likely) … n(roll a 4) =1 • Theoretical probability of an event Ais • Ex: Find the probability of rolling a 4 on a fairdie. P(rolling a 4) • RECALL: 0 < P(A) < 1; if P(A) = 0 then event Ais • impossible but if P(A) = 1 then event A iscertain.

  3. Ex: Find the sample space U for the sex ofthe children (boy/girl) in a family with 2children. List all outcomes in atable: n(U) =4 Find the probability that the two children arebothgirls. Find the probability that the two children are a boy and a girl (not necessarily in thatorder).

  4. Ex: Find the sample space U for the arrangementsof the letters in the word BLUE (4! =24Arrangements) ULBE ELBU ULEB ELUB UELB EULB UEBL EUBL UBLE EBLU UBEL EBUL BBLUE LBUE BBLEU LBEU BULELUBE BUELLUEB BBELU LEBU BBEUL LEUB Find the probability of the arrangement beingan English word. n(u)=24 n(A)=3 P(English)=3/24 =1/8

  5. If the probability of an event is P, in t trialsyou would expect the event to occur tPtimes. Example: A fair 20 sided die with faces numbered 1 to 20isrolled. Find P(rolling a multiple of4) Sample space U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12,13,14, 15, 16, 17, 18,19,20} n(U) =20 n(multiple of 4) =5 P(rolling a multiple of 4) If the die is rolled 100 times, how many times would you expect a multiple of4?

  6. In a standard pack (deck) of 52 playing cards: • There are four different suits : black spades,red • hearts, black clubs, reddiamonds • There are 13 cards in each suit (13 times 4 = 52) • Each suit consists of nine number cards: 2 to 10and • four picture cards: jack, queen, king andace • Ex: find the probability of drawing one card from a deck and it is aspade Ex: if 40 cards are drawn with replacement (the card drawn is put back in the deck each time) how many spades would you expect todraw?

  7. Complement The complement of Event A is denoted byA’ P(A) + P(A’) =1 Complements are mutually exclusive (nooverlap) Example: P(rolling a 4) = 1/6 P(not rolling a 4) =5/6 OR On a fair die find P(rolling a 3 or4) = P(roll 3) + P(roll 4) Draw one card from apack.Find P(heart orace) = P(heart) + P(Ace) – P(ace ofhearts)

  8. Roll two dice. Find the probability of rolling a sum of7 Make achart. n(U) =36 n (A) =6 P(sum of7)

  9. Experimental (empirical)Probability The relative frequency (or empirical probability) of an event is the absolute frequency divided by the total number ofevents You can use relative frequency as an estimate of probability. The larger the number of trials, the closer the relative frequency is to theprobability Ex: ages of students at aHS Find the relativeFrequencies Find P(15 yearsold)= Find P(under 16 years old)= P(13 or 14 or 15 yrsold)= Find P(at least 16 years old)= P(16 or 17 or 18 yrs old)=

  10. Experimental (empirical)Probability Try: The colors of cars passing the school marque yesterday morning are given in thetable. This morning 350 cars pass themarque.Estimatethe number of red cars thismorning. Estimate P(next car that passed wasred)

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