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Odds are ... (Posted on 2017-02-14) Difficulty: 3 of 5
You face an urn with 5555 cards in it, each has a non-zero integer written on it. Nothing is said about the distribution of those numbers. You are told to draw randomly a card, copy the number, return it back, shuffle and draw randomly a card, then write down the sum of both numbers, say S.

(i) Prove: The probability of S being an even number is higher than S being odd.
(ii) Is it true for any initial number of cards? Comment.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Higher numbers before overflow | Comment 5 of 8 |
 The probabilities do get closer to 1/2 as the number of cards gets higher and higher:

 105 53//105 0.5047619047619047619 
 110 111//220 0.5045454545454545454 
 115 58//115 0.5043478260869565217 
 120 121//240 0.5041666666666666666 
 125 63//125 0.504 
 130 131//260 0.5038461538461538461 
 135 68//135 0.5037037037037037036 
 140 141//280 0.5035714285714285714 
 145 73//145 0.5034482758620689654 
 150 151//300 0.5033333333333333333 
 155 78//155 0.5032258064516129032 
 160 161//320 0.503125 
 165 83//165 0.5030303030303030303 
 170 171//340 0.5029411764705882352 
 175 88//175 0.5028571428571428571 
 180 181//360 0.5027777777777777777 
 185 93//185 0.5027027027027027026 
 190 191//380 0.502631578947368421 
 195 98//195 0.5025641025641025641 
 200 201//400 0.5025 
 205 103//205 0.5024390243902439024 
 210 211//420 0.5023809523809523809 
 215 108//215 0.5023255813953488371 
 220 221//440 0.5022727272727272727 
 225 113//225 0.5022222222222222222 
 230 231//460 0.5021739130434782608 
 235 118//235 0.502127659574468085 
 240 241//480 0.5020833333333333333 
 245 123//245 0.5020408163265306122 
 250 251//500 0.502 
 255 128//255 0.5019607843137254901 
 260 261//520 0.501923076923076923 
 265 133//265 0.5018867924528301886 
 270 271//540 0.5018518518518518518 
 275 138//275 0.5018181818181818181 
 280 281//560 0.5017857142857142857 
 285 143//285 0.5017543859649122806 
 290 291//580 0.5017241379310344827 
 295 148//295 0.501694915254237288 
 300 301//600 0.5016666666666666666 
 305 153//305 0.5016393442622950819 
 310 311//620 0.5016129032258064516 
 315 158//315 0.5015873015873015873 
 320 321//640 0.5015625 
 325 163//325 0.5015384615384615384 
 330 331//660 0.5015151515151515151 
 335 168//335 0.5014925373134328357 
 340 341//680 0.5014705882352941176 
 345 173//345 0.5014492753623188405 
 350 351//700 0.5014285714285714285 
 355 178//355 0.501408450704225352 
 360 361//720 0.5013888888888888888 
 365 183//365 0.50136986301369863 
 370 371//740 0.5013513513513513513 
 375 188//375 0.5013333333333333332 
 380 381//760 0.5013157894736842104 
 385 193//385 0.5012987012987012987 
 390 391//780 0.501282051282051282 
 395 198//395 0.5012658227848101265 
 400 401//800 0.50125 
 405 203//405 0.5012345679012345678 
 410 411//820 0.5012195121951219512 
 415 208//415 0.5012048192771084337 
 420 421//840 0.5011904761904761904 
 425 213//425 0.501176470588235294 
 430 431//860 0.5011627906976744185 
 435 218//435 0.5011494252873563218 
 440 441//880 0.5011363636363636363 
 445 223//445 0.5011235955056179775 
 450 451//900 0.5011111111111111111 
 455 228//455 0.5010989010989010989 
 460 461//920 0.5010869565217391303 
 465 233//465 0.501075268817204301 
 470 471//940 0.5010638297872340425 
 475 238//475 0.5010526315789473683 
 480 481//960 0.5010416666666666666 
 485 243//485 0.5010309278350515463 
 490 491//980 0.5010204081632653061 
 495 248//495 0.5010101010101010101 
 500 501//1000 0.501 
 505 253//505 0.50099009900990099 
 510 511//1020 0.500980392156862745 
 515 258//515 0.5009708737864077669 
 520 521//1040 0.5009615384615384615 
 525 263//525 0.5009523809523809523 
 530 531//1060 0.5009433962264150942 
 535 268//535 0.5009345794392523364 
 540 541//1080 0.5009259259259259259 
 545 273//545 0.5009174311926605504 
 550 551//1100 0.500909090909090909 
 555 278//555 0.5009009009009009008 


  Posted by Charlie on 2017-02-14 11:42:28
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