39:
1131:
2175:
798:
1946:
1126:{\displaystyle P({A|C})={\frac {P({A,C})}{P(C)}}={\frac {\sum \limits _{n}{P({A,{B_{n}},C})}}{P(C)}}={\frac {\sum \limits _{n}P({A\mid {B_{n}},C})P({{B_{n}}\mid C})P(C)}{P(C)}}=\sum \limits _{n}P({A\mid {B_{n}},C})P({{B_{n}}\mid C})}
2439:. One author uses the terminology of the "Rule of Average Conditional Probabilities", while another refers to it as the "continuous law of alternatives" in the continuous case. This result is given by Grimmett and Welsh as the
2170:{\displaystyle {\begin{aligned}P(A)&=P(A\mid B_{X})\cdot P(B_{X})+P(A\mid B_{Y})\cdot P(B_{Y})\\&={99 \over 100}\cdot {6 \over 10}+{95 \over 100}\cdot {4 \over 10}={{594+380} \over 1000}={974 \over 1000}\end{aligned}}}
1912:
1734:
1579:
1313:
1951:
448:
644:
2407:
2346:
549:
1476:
1363:
2285:
2230:
751:
1774:
1937:
supplies 60% of the total bulbs available and Y supplies 40% of the total bulbs available. What is the chance that a purchased bulb will work for longer than 5000 hours?
709:
1628:
1414:
1212:
1161:
787:
1794:
1601:
1434:
1387:
1181:
667:
480:
330:
1801:
1635:
1483:
1220:
323:
2619:
2565:
2535:
386:
560:
1323:
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let
2582:
2592:
2511:
1184:
316:
304:
263:
2357:
2296:
487:
194:
130:
1439:
1326:
242:
103:
789:, is sometimes called "average probability"; "overall probability" is sometimes used in less formal writings.
2697:
2241:
2186:
368:
98:
2702:
364:
214:
714:
2461:
2444:
2436:
268:
157:
142:
1741:
2471:
459:
252:
123:
672:
2476:
147:
2609:
367:. It expresses the total probability of an outcome which can be realized via several distinct
2555:
2525:
2481:
2466:
288:
247:
152:
118:
2456:
1606:
1392:
1190:
1139:
360:
278:
172:
65:
763:
8:
237:
179:
167:
162:
1779:
1586:
1419:
1372:
1166:
652:
465:
455:
344:
224:
113:
53:
30:
2638:
2615:
2588:
2561:
2551:
2531:
2507:
1366:
283:
189:
88:
2417:
Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.
757:
108:
38:
2672:, by John J. Schiller, Seymour Lipschutz, McGraw–Hill Professional, 2010, page 89.
184:
135:
792:
The law of total probability can also be stated for conditional probabilities:
199:
2691:
2642:
451:
72:
2660:
by Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
1933:'s bulbs work for over 5000 hours in 95% of cases. It is known that factory
1907:{\displaystyle P(Y\in B)=\int _{-\infty }^{\infty }P(Y\in B|X=x)f_{X}(x)dx.}
299:
209:
93:
219:
60:
48:
2287:
is the probability that the purchased bulb was manufactured by factory
2232:
is the probability that the purchased bulb was manufactured by factory
1922:
77:
23:
2435:, which is a special case of the law of total probability applying to
1929:'s bulbs work for over 5000 hours in 99% of cases, whereas factory
204:
1729:{\displaystyle P(A)=\int _{-\infty }^{\infty }P(A|X=x)f_{X}(x)dx.}
1574:{\displaystyle P(A)=\int _{-\infty }^{\infty }P(A|X=x)dF_{X}(x).}
380:
711:, then these terms are simply omitted from the summation since
1308:{\displaystyle P(A\mid C)=\sum _{n}P(A\mid C,B_{n})P(B_{n})}
2678:, by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
2504:
CRC Standard
Probability and Statistics Tables and Formulae
443:{\displaystyle \left\{{B_{n}:n=1,2,3,\ldots }\right\}}
2360:
2299:
2244:
2189:
1949:
1804:
1782:
1744:
1638:
1609:
1589:
1486:
1442:
1422:
1395:
1375:
1329:
1223:
1193:
1169:
1142:
801:
766:
717:
675:
655:
639:{\displaystyle P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),}
563:
490:
468:
389:
2401:
2340:
2279:
2224:
2169:
1906:
1788:
1768:
1728:
1622:
1595:
1573:
1470:
1428:
1408:
1381:
1357:
1307:
1206:
1175:
1155:
1125:
781:
745:
703:
661:
638:
543:
474:
442:
2607:
2689:
2645:, Oxford Science Publications, 1986, Theorem 1B.
2523:
1940:Applying the law of total probability, we have:
1389:is a random variable with distribution function
2670:Schaum's Outline of Probability, Second Edition
2409:is the probability that a bulb manufactured by
2348:is the probability that a bulb manufactured by
2550:
2530:. Courier Dover Publications. pp. 47–48.
2402:{\displaystyle P(A\mid B_{Y})={95 \over 100}}
2341:{\displaystyle P(A\mid B_{X})={99 \over 100}}
760:, and consequently the marginal probability,
324:
2601:
2443:, a name that they also give to the related
1763:
1751:
1478:. Then the law of total probability states
544:{\displaystyle P(A)=\sum _{n}P(A\cap B_{n})}
2517:
2658:Introduction to Probability and Statistics
2580:
1471:{\displaystyle (\Omega ,{\mathcal {F}},P)}
1358:{\displaystyle (\Omega ,{\mathcal {F}},P)}
331:
317:
2684:, by Alan Gut, Springer, 1995, pages 5–6.
2574:
2544:
2666:, by Mark J. Schervish, Springer, 1995.
2690:
1738:Moreover, for the specific case where
756:The summation can be interpreted as a
383:that states, in its discrete case, if
2682:An Intermediate Course in Probability
2280:{\displaystyle P(B_{Y})={4 \over 10}}
2225:{\displaystyle P(B_{X})={6 \over 10}}
2498:
2496:
2676:A First Course in Stochastic Models
2502:Zwillinger, D., Kokoska, S. (2000)
1050:
944:
874:
13:
2611:Introduction to probability with R
1921:Suppose that two factories supply
1839:
1834:
1796:is a Borel set, then this yields
1667:
1662:
1515:
1510:
1454:
1446:
1341:
1333:
1318:
379:The law of total probability is a
14:
2714:
2493:
37:
2431:is sometimes taken to mean the
359:is a fundamental rule relating
2628:
2527:Concepts of probability theory
2420:
2413:will work for over 5000 hours.
2383:
2364:
2352:will work for over 5000 hours;
2322:
2303:
2261:
2248:
2206:
2193:
2067:
2054:
2045:
2026:
2017:
2004:
1995:
1976:
1963:
1957:
1892:
1886:
1873:
1860:
1847:
1820:
1808:
1714:
1708:
1695:
1682:
1675:
1648:
1642:
1565:
1559:
1543:
1530:
1523:
1496:
1490:
1465:
1443:
1352:
1330:
1302:
1289:
1283:
1258:
1239:
1227:
1120:
1097:
1091:
1062:
1040:
1034:
1026:
1020:
1014:
991:
985:
956:
931:
925:
916:
887:
861:
855:
847:
833:
821:
813:
805:
776:
770:
746:{\displaystyle P(A\mid B_{n})}
740:
721:
692:
679:
630:
617:
611:
592:
573:
567:
538:
519:
500:
494:
104:Collectively exhaustive events
1:
2651:
16:Concept in probability theory
2635:Probability: An Introduction
1769:{\displaystyle A=\{Y\in B\}}
374:
7:
2560:. For Dummies. p. 58.
2450:
462:events, then for any event
10:
2719:
2614:. CRC Press. p. 179.
2608:Kenneth Baclawski (2008).
1916:
1603:admits a density function
704:{\displaystyle P(B_{n})=0}
2524:Paul E. Pfeiffer (1978).
2437:discrete random variables
365:conditional probabilities
2587:. Springer. p. 41.
2487:
2462:Law of total expectation
2445:law of total expectation
2428:law of total probability
274:Law of total probability
269:Conditional independence
158:Exponential distribution
143:Probability distribution
2557:Probability for dummies
2472:Law of total covariance
1925:to the market. Factory
1163:as above, and assuming
460:collectively exhaustive
253:Conditional probability
2477:Law of total cumulance
2403:
2342:
2281:
2226:
2171:
1908:
1790:
1770:
1730:
1630:, then the result is
1624:
1597:
1575:
1472:
1430:
1410:
1383:
1359:
1309:
1208:
1177:
1157:
1127:
783:
747:
705:
663:
640:
545:
476:
444:
361:marginal probabilities
195:Continuous or discrete
148:Bernoulli distribution
2482:Marginal distribution
2467:Law of total variance
2404:
2343:
2282:
2227:
2172:
1909:
1791:
1771:
1731:
1625:
1623:{\displaystyle f_{X}}
1598:
1576:
1473:
1431:
1411:
1409:{\displaystyle F_{X}}
1384:
1360:
1310:
1209:
1207:{\displaystyle B_{n}}
1178:
1158:
1156:{\displaystyle B_{n}}
1128:
784:
748:
706:
664:
641:
546:
477:
445:
153:Binomial distribution
2698:Probability theorems
2664:Theory of Statistics
2457:Law of large numbers
2358:
2297:
2242:
2187:
1947:
1802:
1780:
1742:
1636:
1607:
1587:
1484:
1440:
1420:
1393:
1373:
1327:
1221:
1191:
1167:
1140:
799:
782:{\displaystyle P(A)}
764:
715:
673:
653:
561:
488:
466:
387:
357:of total probability
279:Law of large numbers
248:Marginal probability
173:Poisson distribution
22:Part of a series on
2581:Jim Pitman (1993).
2433:law of alternatives
1843:
1671:
1519:
554:or, alternatively,
238:Complementary event
180:Probability measure
168:Pareto distribution
163:Normal distribution
2399:
2338:
2277:
2222:
2167:
2165:
1904:
1826:
1786:
1766:
1726:
1654:
1620:
1593:
1571:
1502:
1468:
1426:
1406:
1379:
1355:
1305:
1254:
1204:
1173:
1153:
1123:
1058:
952:
882:
779:
743:
701:
659:
636:
588:
541:
515:
472:
456:mutually exclusive
452:countably infinite
440:
371:, hence the name.
345:probability theory
289:Boole's inequality
225:Stochastic process
114:Mutual exclusivity
31:Probability theory
2639:Geoffrey Grimmett
2621:978-1-4200-6521-3
2567:978-0-471-75141-0
2537:978-0-486-63677-1
2441:partition theorem
2397:
2336:
2275:
2220:
2161:
2148:
2127:
2114:
2101:
2088:
1789:{\displaystyle B}
1596:{\displaystyle X}
1429:{\displaystyle A}
1382:{\displaystyle X}
1367:probability space
1245:
1176:{\displaystyle C}
1049:
1044:
943:
935:
873:
865:
662:{\displaystyle n}
579:
506:
475:{\displaystyle A}
341:
340:
243:Joint probability
190:Bernoulli process
89:Probability space
2710:
2703:Statistical laws
2646:
2632:
2626:
2625:
2605:
2599:
2598:
2578:
2572:
2571:
2548:
2542:
2541:
2521:
2515:
2500:
2408:
2406:
2405:
2400:
2398:
2390:
2382:
2381:
2347:
2345:
2344:
2339:
2337:
2329:
2321:
2320:
2286:
2284:
2283:
2278:
2276:
2268:
2260:
2259:
2231:
2229:
2228:
2223:
2221:
2213:
2205:
2204:
2176:
2174:
2173:
2168:
2166:
2162:
2154:
2149:
2144:
2133:
2128:
2120:
2115:
2107:
2102:
2094:
2089:
2081:
2073:
2066:
2065:
2044:
2043:
2016:
2015:
1994:
1993:
1913:
1911:
1910:
1905:
1885:
1884:
1863:
1842:
1837:
1795:
1793:
1792:
1787:
1775:
1773:
1772:
1767:
1735:
1733:
1732:
1727:
1707:
1706:
1685:
1670:
1665:
1629:
1627:
1626:
1621:
1619:
1618:
1602:
1600:
1599:
1594:
1580:
1578:
1577:
1572:
1558:
1557:
1533:
1518:
1513:
1477:
1475:
1474:
1469:
1458:
1457:
1435:
1433:
1432:
1427:
1415:
1413:
1412:
1407:
1405:
1404:
1388:
1386:
1385:
1380:
1364:
1362:
1361:
1356:
1345:
1344:
1314:
1312:
1311:
1306:
1301:
1300:
1282:
1281:
1253:
1213:
1211:
1210:
1205:
1203:
1202:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1152:
1151:
1132:
1130:
1129:
1124:
1119:
1112:
1111:
1110:
1090:
1083:
1082:
1081:
1057:
1045:
1043:
1029:
1013:
1006:
1005:
1004:
984:
977:
976:
975:
951:
941:
936:
934:
920:
919:
915:
908:
907:
906:
881:
871:
866:
864:
850:
846:
828:
820:
816:
788:
786:
785:
780:
758:weighted average
752:
750:
749:
744:
739:
738:
710:
708:
707:
702:
691:
690:
668:
666:
665:
660:
645:
643:
642:
637:
629:
628:
610:
609:
587:
550:
548:
547:
542:
537:
536:
514:
481:
479:
478:
473:
449:
447:
446:
441:
439:
435:
404:
403:
333:
326:
319:
109:Elementary event
41:
19:
18:
2718:
2717:
2713:
2712:
2711:
2709:
2708:
2707:
2688:
2687:
2654:
2649:
2633:
2629:
2622:
2606:
2602:
2595:
2579:
2575:
2568:
2549:
2545:
2538:
2522:
2518:
2501:
2494:
2490:
2453:
2423:
2389:
2377:
2373:
2359:
2356:
2355:
2328:
2316:
2312:
2298:
2295:
2294:
2267:
2255:
2251:
2243:
2240:
2239:
2212:
2200:
2196:
2188:
2185:
2184:
2164:
2163:
2153:
2134:
2132:
2119:
2106:
2093:
2080:
2071:
2070:
2061:
2057:
2039:
2035:
2011:
2007:
1989:
1985:
1966:
1950:
1948:
1945:
1944:
1919:
1880:
1876:
1859:
1838:
1830:
1803:
1800:
1799:
1781:
1778:
1777:
1743:
1740:
1739:
1702:
1698:
1681:
1666:
1658:
1637:
1634:
1633:
1614:
1610:
1608:
1605:
1604:
1588:
1585:
1584:
1553:
1549:
1529:
1514:
1506:
1485:
1482:
1481:
1453:
1452:
1441:
1438:
1437:
1421:
1418:
1417:
1400:
1396:
1394:
1391:
1390:
1374:
1371:
1370:
1340:
1339:
1328:
1325:
1324:
1321:
1319:Continuous case
1296:
1292:
1277:
1273:
1249:
1222:
1219:
1218:
1198:
1194:
1192:
1189:
1188:
1168:
1165:
1164:
1147:
1143:
1141:
1138:
1137:
1106:
1102:
1101:
1100:
1077:
1073:
1072:
1065:
1053:
1030:
1000:
996:
995:
994:
971:
967:
966:
959:
947:
942:
940:
921:
902:
898:
897:
890:
883:
877:
872:
870:
851:
836:
829:
827:
812:
808:
800:
797:
796:
765:
762:
761:
734:
730:
716:
713:
712:
686:
682:
674:
671:
670:
654:
651:
650:
649:where, for any
624:
620:
605:
601:
583:
562:
559:
558:
532:
528:
510:
489:
486:
485:
467:
464:
463:
450:is a finite or
399:
395:
394:
390:
388:
385:
384:
377:
337:
185:Random variable
136:Bernoulli trial
17:
12:
11:
5:
2716:
2706:
2705:
2700:
2686:
2685:
2679:
2673:
2667:
2661:
2653:
2650:
2648:
2647:
2627:
2620:
2600:
2593:
2573:
2566:
2552:Deborah Rumsey
2543:
2536:
2516:
2491:
2489:
2486:
2485:
2484:
2479:
2474:
2469:
2464:
2459:
2452:
2449:
2422:
2419:
2415:
2414:
2396:
2393:
2388:
2385:
2380:
2376:
2372:
2369:
2366:
2363:
2353:
2335:
2332:
2327:
2324:
2319:
2315:
2311:
2308:
2305:
2302:
2292:
2274:
2271:
2266:
2263:
2258:
2254:
2250:
2247:
2237:
2219:
2216:
2211:
2208:
2203:
2199:
2195:
2192:
2178:
2177:
2160:
2157:
2152:
2147:
2143:
2140:
2137:
2131:
2126:
2123:
2118:
2113:
2110:
2105:
2100:
2097:
2092:
2087:
2084:
2079:
2076:
2074:
2072:
2069:
2064:
2060:
2056:
2053:
2050:
2047:
2042:
2038:
2034:
2031:
2028:
2025:
2022:
2019:
2014:
2010:
2006:
2003:
2000:
1997:
1992:
1988:
1984:
1981:
1978:
1975:
1972:
1969:
1967:
1965:
1962:
1959:
1956:
1953:
1952:
1918:
1915:
1903:
1900:
1897:
1894:
1891:
1888:
1883:
1879:
1875:
1872:
1869:
1866:
1862:
1858:
1855:
1852:
1849:
1846:
1841:
1836:
1833:
1829:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1785:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1725:
1722:
1719:
1716:
1713:
1710:
1705:
1701:
1697:
1694:
1691:
1688:
1684:
1680:
1677:
1674:
1669:
1664:
1661:
1657:
1653:
1650:
1647:
1644:
1641:
1617:
1613:
1592:
1570:
1567:
1564:
1561:
1556:
1552:
1548:
1545:
1542:
1539:
1536:
1532:
1528:
1525:
1522:
1517:
1512:
1509:
1505:
1501:
1498:
1495:
1492:
1489:
1467:
1464:
1461:
1456:
1451:
1448:
1445:
1425:
1403:
1399:
1378:
1354:
1351:
1348:
1343:
1338:
1335:
1332:
1320:
1317:
1316:
1315:
1304:
1299:
1295:
1291:
1288:
1285:
1280:
1276:
1272:
1269:
1266:
1263:
1260:
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1252:
1248:
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1235:
1232:
1229:
1226:
1201:
1197:
1187:of any of the
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832:
826:
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811:
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778:
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772:
769:
742:
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572:
569:
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284:Bayes' theorem
281:
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257:
256:
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232:
231:
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215:Observed value
212:
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200:Expected value
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34:
33:
27:
26:
15:
9:
6:
4:
3:
2:
2715:
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2655:
2644:
2643:Dominic Welsh
2640:
2636:
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2613:
2612:
2604:
2596:
2594:0-387-97974-3
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2512:1-58488-059-7
2509:
2506:, CRC Press.
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2012:
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1998:
1990:
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1199:
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1103:
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731:
727:
724:
718:
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683:
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656:
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625:
621:
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602:
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595:
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491:
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100:
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84:
79:
76:
74:
73:Indeterminism
71:
67:
64:
63:
62:
59:
55:
52:
51:
50:
47:
46:
45:
44:
40:
36:
35:
32:
29:
28:
25:
21:
20:
2681:
2675:
2669:
2663:
2657:
2634:
2630:
2610:
2603:
2583:
2576:
2556:
2546:
2526:
2519:
2503:
2440:
2432:
2427:
2426:
2424:
2416:
2410:
2349:
2288:
2233:
2179:
1939:
1934:
1930:
1926:
1920:
1798:
1737:
1632:
1582:
1480:
1436:an event on
1322:
1183:is an event
1135:
791:
755:
648:
553:
378:
356:
352:
348:
342:
305:Tree diagram
300:Venn diagram
273:
264:Independence
210:Markov chain
94:Sample space
2584:Probability
2421:Other names
1923:light bulbs
1185:independent
1136:Taking the
753:is finite.
220:Random walk
61:Determinism
49:Probability
2692:Categories
2652:References
1369:. Suppose
131:Experiment
78:Randomness
24:statistics
2425:The term
2371:∣
2310:∣
2117:⋅
2091:⋅
2049:⋅
2033:∣
1999:⋅
1983:∣
1854:∈
1840:∞
1835:∞
1832:−
1828:∫
1815:∈
1758:∈
1668:∞
1663:∞
1660:−
1656:∫
1516:∞
1511:∞
1508:−
1504:∫
1447:Ω
1334:Ω
1265:∣
1247:∑
1234:∣
1114:∣
1070:∣
1051:∑
1008:∣
964:∣
945:∑
875:∑
728:∣
599:∣
581:∑
526:∩
508:∑
433:…
375:Statement
124:Singleton
2554:(2006).
2514:page 31.
2451:See also
1776:, where
205:Variance
1917:Example
454:set of
381:theorem
353:formula
119:Outcome
2618:
2591:
2564:
2534:
2510:
2180:where
1416:, and
369:events
347:, the
66:System
54:Axioms
2637:, by
2488:Notes
1365:be a
669:, if
99:Event
2641:and
2616:ISBN
2589:ISBN
2562:ISBN
2532:ISBN
2508:ISBN
2159:1000
2146:1000
458:and
351:(or
2395:100
2334:100
2156:974
2142:380
2136:594
2112:100
2086:100
1583:If
363:to
349:law
343:In
2694::
2495:^
2447:.
2392:95
2331:99
2273:10
2218:10
2125:10
2109:95
2099:10
2083:99
1214::
355:)
2624:.
2597:.
2570:.
2540:.
2411:Y
2387:=
2384:)
2379:Y
2375:B
2368:A
2365:(
2362:P
2350:X
2326:=
2323:)
2318:X
2314:B
2307:A
2304:(
2301:P
2291:;
2289:Y
2270:4
2265:=
2262:)
2257:Y
2253:B
2249:(
2246:P
2236:;
2234:X
2215:6
2210:=
2207:)
2202:X
2198:B
2194:(
2191:P
2151:=
2139:+
2130:=
2122:4
2104:+
2096:6
2078:=
2068:)
2063:Y
2059:B
2055:(
2052:P
2046:)
2041:Y
2037:B
2030:A
2027:(
2024:P
2021:+
2018:)
2013:X
2009:B
2005:(
2002:P
1996:)
1991:X
1987:B
1980:A
1977:(
1974:P
1971:=
1964:)
1961:A
1958:(
1955:P
1935:X
1931:Y
1927:X
1902:.
1899:x
1896:d
1893:)
1890:x
1887:(
1882:X
1878:f
1874:)
1871:x
1868:=
1865:X
1861:|
1857:B
1851:Y
1848:(
1845:P
1824:=
1821:)
1818:B
1812:Y
1809:(
1806:P
1784:B
1764:}
1761:B
1755:Y
1752:{
1749:=
1746:A
1724:.
1721:x
1718:d
1715:)
1712:x
1709:(
1704:X
1700:f
1696:)
1693:x
1690:=
1687:X
1683:|
1679:A
1676:(
1673:P
1652:=
1649:)
1646:A
1643:(
1640:P
1616:X
1612:f
1591:X
1569:.
1566:)
1563:x
1560:(
1555:X
1551:F
1547:d
1544:)
1541:x
1538:=
1535:X
1531:|
1527:A
1524:(
1521:P
1500:=
1497:)
1494:A
1491:(
1488:P
1466:)
1463:P
1460:,
1455:F
1450:,
1444:(
1424:A
1402:X
1398:F
1377:X
1353:)
1350:P
1347:,
1342:F
1337:,
1331:(
1303:)
1298:n
1294:B
1290:(
1287:P
1284:)
1279:n
1275:B
1271:,
1268:C
1262:A
1259:(
1256:P
1251:n
1243:=
1240:)
1237:C
1231:A
1228:(
1225:P
1200:n
1196:B
1171:C
1149:n
1145:B
1121:)
1117:C
1108:n
1104:B
1098:(
1095:P
1092:)
1088:C
1085:,
1079:n
1075:B
1067:A
1063:(
1060:P
1055:n
1047:=
1041:)
1038:C
1035:(
1032:P
1027:)
1024:C
1021:(
1018:P
1015:)
1011:C
1002:n
998:B
992:(
989:P
986:)
982:C
979:,
973:n
969:B
961:A
957:(
954:P
949:n
938:=
932:)
929:C
926:(
923:P
917:)
913:C
910:,
904:n
900:B
895:,
892:A
888:(
885:P
879:n
868:=
862:)
859:C
856:(
853:P
848:)
844:C
841:,
838:A
834:(
831:P
825:=
822:)
818:C
814:|
810:A
806:(
803:P
777:)
774:A
771:(
768:P
741:)
736:n
732:B
725:A
722:(
719:P
699:0
696:=
693:)
688:n
684:B
680:(
677:P
657:n
634:,
631:)
626:n
622:B
618:(
615:P
612:)
607:n
603:B
596:A
593:(
590:P
585:n
577:=
574:)
571:A
568:(
565:P
539:)
534:n
530:B
523:A
520:(
517:P
512:n
504:=
501:)
498:A
495:(
492:P
470:A
437:}
430:,
427:3
424:,
421:2
418:,
415:1
412:=
409:n
406::
401:n
397:B
392:{
332:e
325:t
318:v
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