Gretchen and Henry were sent to their rooms for fighting in the house. They each separately voiced their protest to their father, insisting that the fight was nothing more than healthy sibling competition, and they each wanted to go out that afternoon to see a movie. He was moved by their stories, but wouldn't simply set them free. Instead, he devised a system. He went to each child's room with a penny, and told them that they would have to show up in the den in 10 minutes, and could choose to bring the penny with them or leave it in their respective rooms. Dad would then flip the one or two pennies brought to the den, and if the pennies he flipped came up heads, the kids could go to the movies. If neither brought a penny, or if he flipped at least one tail, they would stay in their rooms until supper time.
The problem facing Gretchen and Henry was that neither knew what the other would do. It would be easy if they could collude -- one would bring a penny, and the other would not, giving them a 50% chance of going free -- but they did not have this luxury.
If they both acted optimally, what is the probability that they will be free in time to see the movie?
If G and H merely tossed their coin to determine if he or she would bring the coin downstairs, each would have a 1/2 probability of bringing one, and the probability of winning would be .5^2 + .25/4 = .25 + .0625 = .3125.
Since having two pennies is preferable to having no pennies, it could be best to have a higher than 1/2 probability of bringing one's own penny, but how much higher?
Individually each one must determine the best probability of bringing down his or her penny. Any scheme to randomize what the probability should be, would still result ultimately in one probability altogether, and so would be a futile complication.
The table below shows that clearly be best probability that can be achieved, of seeing the movie, would be if the probability of bringing the penny, for each one independently is made to be 2/3, giving them a 1/3 probability of seeing the movie.
They could each devise some way of flipping the coin a number of times to approximate a 2/3 probability. Or, if they are lucky enough to have a 12-sided die in their room to select 8 sides that showing up would indicate to take the penny downstairs, that could be done. Or, childrens' games often include a spinner, and pencil markings could be made on it to mark off 2/3 of the circumference to designate "bring the penny".
But to answer the question, the probability of seeing the movie, under the best strategy, is 1/3.
prob. probability
of of seeing
bringing movie
penny
0.100 0.09250000
0.101 0.09334925
0.102 0.09419700
0.103 0.09504325
0.104 0.09588800
0.105 0.09673125
0.106 0.09757300
0.107 0.09841325
0.108 0.09925200
0.109 0.10008925
0.110 0.10092500
0.111 0.10175925
0.112 0.10259200
0.113 0.10342325
0.114 0.10425300
0.115 0.10508125
0.116 0.10590800
0.117 0.10673325
0.118 0.10755700
0.119 0.10837925
0.120 0.10920000
0.121 0.11001925
0.122 0.11083700
0.123 0.11165325
0.124 0.11246800
0.125 0.11328125
0.126 0.11409300
0.127 0.11490325
0.128 0.11571200
0.129 0.11651925
0.130 0.11732500
0.131 0.11812925
0.132 0.11893200
0.133 0.11973325
0.134 0.12053300
0.135 0.12133125
0.136 0.12212800
0.137 0.12292325
0.138 0.12371700
0.139 0.12450925
0.140 0.12530000
0.141 0.12608925
0.142 0.12687700
0.143 0.12766325
0.144 0.12844800
0.145 0.12923125
0.146 0.13001300
0.147 0.13079325
0.148 0.13157200
0.149 0.13234925
0.150 0.13312500
0.151 0.13389925
0.152 0.13467200
0.153 0.13544325
0.154 0.13621300
0.155 0.13698125
0.156 0.13774800
0.157 0.13851325
0.158 0.13927700
0.159 0.14003925
0.160 0.14080000
0.161 0.14155925
0.162 0.14231700
0.163 0.14307325
0.164 0.14382800
0.165 0.14458125
0.166 0.14533300
0.167 0.14608325
0.168 0.14683200
0.169 0.14757925
0.170 0.14832500
0.171 0.14906925
0.172 0.14981200
0.173 0.15055325
0.174 0.15129300
0.175 0.15203125
0.176 0.15276800
0.177 0.15350325
0.178 0.15423700
0.179 0.15496925
0.180 0.15570000
0.181 0.15642925
0.182 0.15715700
0.183 0.15788325
0.184 0.15860800
0.185 0.15933125
0.186 0.16005300
0.187 0.16077325
0.188 0.16149200
0.189 0.16220925
0.190 0.16292500
0.191 0.16363925
0.192 0.16435200
0.193 0.16506325
0.194 0.16577300
0.195 0.16648125
0.196 0.16718800
0.197 0.16789325
0.198 0.16859700
0.199 0.16929925
0.200 0.17000000
0.201 0.17069925
0.202 0.17139700
0.203 0.17209325
0.204 0.17278800
0.205 0.17348125
0.206 0.17417300
0.207 0.17486325
0.208 0.17555200
0.209 0.17623925
0.210 0.17692500
0.211 0.17760925
0.212 0.17829200
0.213 0.17897325
0.214 0.17965300
0.215 0.18033125
0.216 0.18100800
0.217 0.18168325
0.218 0.18235700
0.219 0.18302925
0.220 0.18370000
0.221 0.18436925
0.222 0.18503700
0.223 0.18570325
0.224 0.18636800
0.225 0.18703125
0.226 0.18769300
0.227 0.18835325
0.228 0.18901200
0.229 0.18966925
0.230 0.19032500
0.231 0.19097925
0.232 0.19163200
0.233 0.19228325
0.234 0.19293300
0.235 0.19358125
0.236 0.19422800
0.237 0.19487325
0.238 0.19551700
0.239 0.19615925
0.240 0.19680000
0.241 0.19743925
0.242 0.19807700
0.243 0.19871325
0.244 0.19934800
0.245 0.19998125
0.246 0.20061300
0.247 0.20124325
0.248 0.20187200
0.249 0.20249925
0.250 0.20312500
0.251 0.20374925
0.252 0.20437200
0.253 0.20499325
0.254 0.20561300
0.255 0.20623125
0.256 0.20684800
0.257 0.20746325
0.258 0.20807700
0.259 0.20868925
0.260 0.20930000
0.261 0.20990925
0.262 0.21051700
0.263 0.21112325
0.264 0.21172800
0.265 0.21233125
0.266 0.21293300
0.267 0.21353325
0.268 0.21413200
0.269 0.21472925
0.270 0.21532500
0.271 0.21591925
0.272 0.21651200
0.273 0.21710325
0.274 0.21769300
0.275 0.21828125
0.276 0.21886800
0.277 0.21945325
0.278 0.22003700
0.279 0.22061925
0.280 0.22120000
0.281 0.22177925
0.282 0.22235700
0.283 0.22293325
0.284 0.22350800
0.285 0.22408125
0.286 0.22465300
0.287 0.22522325
0.288 0.22579200
0.289 0.22635925
0.290 0.22692500
0.291 0.22748925
0.292 0.22805200
0.293 0.22861325
0.294 0.22917300
0.295 0.22973125
0.296 0.23028800
0.297 0.23084325
0.298 0.23139700
0.299 0.23194925
0.300 0.23250000
0.301 0.23304925
0.302 0.23359700
0.303 0.23414325
0.304 0.23468800
0.305 0.23523125
0.306 0.23577300
0.307 0.23631325
0.308 0.23685200
0.309 0.23738925
0.310 0.23792500
0.311 0.23845925
0.312 0.23899200
0.313 0.23952325
0.314 0.24005300
0.315 0.24058125
0.316 0.24110800
0.317 0.24163325
0.318 0.24215700
0.319 0.24267925
0.320 0.24320000
0.321 0.24371925
0.322 0.24423700
0.323 0.24475325
0.324 0.24526800
0.325 0.24578125
0.326 0.24629300
0.327 0.24680325
0.328 0.24731200
0.329 0.24781925
0.330 0.24832500
0.331 0.24882925
0.332 0.24933200
0.333 0.24983325
0.334 0.25033300
0.335 0.25083125
0.336 0.25132800
0.337 0.25182325
0.338 0.25231700
0.339 0.25280925
0.340 0.25330000
0.341 0.25378925
0.342 0.25427700
0.343 0.25476325
0.344 0.25524800
0.345 0.25573125
0.346 0.25621300
0.347 0.25669325
0.348 0.25717200
0.349 0.25764925
0.350 0.25812500
0.351 0.25859925
0.352 0.25907200
0.353 0.25954325
0.354 0.26001300
0.355 0.26048125
0.356 0.26094800
0.357 0.26141325
0.358 0.26187700
0.359 0.26233925
0.360 0.26280000
0.361 0.26325925
0.362 0.26371700
0.363 0.26417325
0.364 0.26462800
0.365 0.26508125
0.366 0.26553300
0.367 0.26598325
0.368 0.26643200
0.369 0.26687925
0.370 0.26732500
0.371 0.26776925
0.372 0.26821200
0.373 0.26865325
0.374 0.26909300
0.375 0.26953125
0.376 0.26996800
0.377 0.27040325
0.378 0.27083700
0.379 0.27126925
0.380 0.27170000
0.381 0.27212925
0.382 0.27255700
0.383 0.27298325
0.384 0.27340800
0.385 0.27383125
0.386 0.27425300
0.387 0.27467325
0.388 0.27509200
0.389 0.27550925
0.390 0.27592500
0.391 0.27633925
0.392 0.27675200
0.393 0.27716325
0.394 0.27757300
0.395 0.27798125
0.396 0.27838800
0.397 0.27879325
0.398 0.27919700
0.399 0.27959925
0.400 0.28000000
0.401 0.28039925
0.402 0.28079700
0.403 0.28119325
0.404 0.28158800
0.405 0.28198125
0.406 0.28237300
0.407 0.28276325
0.408 0.28315200
0.409 0.28353925
0.410 0.28392500
0.411 0.28430925
0.412 0.28469200
0.413 0.28507325
0.414 0.28545300
0.415 0.28583125
0.416 0.28620800
0.417 0.28658325
0.418 0.28695700
0.419 0.28732925
0.420 0.28770000
0.421 0.28806925
0.422 0.28843700
0.423 0.28880325
0.424 0.28916800
0.425 0.28953125
0.426 0.28989300
0.427 0.29025325
0.428 0.29061200
0.429 0.29096925
0.430 0.29132500
0.431 0.29167925
0.432 0.29203200
0.433 0.29238325
0.434 0.29273300
0.435 0.29308125
0.436 0.29342800
0.437 0.29377325
0.438 0.29411700
0.439 0.29445925
0.440 0.29480000
0.441 0.29513925
0.442 0.29547700
0.443 0.29581325
0.444 0.29614800
0.445 0.29648125
0.446 0.29681300
0.447 0.29714325
0.448 0.29747200
0.449 0.29779925
0.450 0.29812500
0.451 0.29844925
0.452 0.29877200
0.453 0.29909325
0.454 0.29941300
0.455 0.29973125
0.456 0.30004800
0.457 0.30036325
0.458 0.30067700
0.459 0.30098925
0.460 0.30130000
0.461 0.30160925
0.462 0.30191700
0.463 0.30222325
0.464 0.30252800
0.465 0.30283125
0.466 0.30313300
0.467 0.30343325
0.468 0.30373200
0.469 0.30402925
0.470 0.30432500
0.471 0.30461925
0.472 0.30491200
0.473 0.30520325
0.474 0.30549300
0.475 0.30578125
0.476 0.30606800
0.477 0.30635325
0.478 0.30663700
0.479 0.30691925
0.480 0.30720000
0.481 0.30747925
0.482 0.30775700
0.483 0.30803325
0.484 0.30830800
0.485 0.30858125
0.486 0.30885300
0.487 0.30912325
0.488 0.30939200
0.489 0.30965925
0.490 0.30992500
0.491 0.31018925
0.492 0.31045200
0.493 0.31071325
0.494 0.31097300
0.495 0.31123125
0.496 0.31148800
0.497 0.31174325
0.498 0.31199700
0.499 0.31224925
0.500 0.31250000
0.501 0.31274925
0.502 0.31299700
0.503 0.31324325
0.504 0.31348800
0.505 0.31373125
0.506 0.31397300
0.507 0.31421325
0.508 0.31445200
0.509 0.31468925
0.510 0.31492500
0.511 0.31515925
0.512 0.31539200
0.513 0.31562325
0.514 0.31585300
0.515 0.31608125
0.516 0.31630800
0.517 0.31653325
0.518 0.31675700
0.519 0.31697925
0.520 0.31720000
0.521 0.31741925
0.522 0.31763700
0.523 0.31785325
0.524 0.31806800
0.525 0.31828125
0.526 0.31849300
0.527 0.31870325
0.528 0.31891200
0.529 0.31911925
0.530 0.31932500
0.531 0.31952925
0.532 0.31973200
0.533 0.31993325
0.534 0.32013300
0.535 0.32033125
0.536 0.32052800
0.537 0.32072325
0.538 0.32091700
0.539 0.32110925
0.540 0.32130000
0.541 0.32148925
0.542 0.32167700
0.543 0.32186325
0.544 0.32204800
0.545 0.32223125
0.546 0.32241300
0.547 0.32259325
0.548 0.32277200
0.549 0.32294925
0.550 0.32312500
0.551 0.32329925
0.552 0.32347200
0.553 0.32364325
0.554 0.32381300
0.555 0.32398125
0.556 0.32414800
0.557 0.32431325
0.558 0.32447700
0.559 0.32463925
0.560 0.32480000
0.561 0.32495925
0.562 0.32511700
0.563 0.32527325
0.564 0.32542800
0.565 0.32558125
0.566 0.32573300
0.567 0.32588325
0.568 0.32603200
0.569 0.32617925
0.570 0.32632500
0.571 0.32646925
0.572 0.32661200
0.573 0.32675325
0.574 0.32689300
0.575 0.32703125
0.576 0.32716800
0.577 0.32730325
0.578 0.32743700
0.579 0.32756925
0.580 0.32770000
0.581 0.32782925
0.582 0.32795700
0.583 0.32808325
0.584 0.32820800
0.585 0.32833125
0.586 0.32845300
0.587 0.32857325
0.588 0.32869200
0.589 0.32880925
0.590 0.32892500
0.591 0.32903925
0.592 0.32915200
0.593 0.32926325
0.594 0.32937300
0.595 0.32948125
0.596 0.32958800
0.597 0.32969325
0.598 0.32979700
0.599 0.32989925
0.600 0.33000000
0.601 0.33009925
0.602 0.33019700
0.603 0.33029325
0.604 0.33038800
0.605 0.33048125
0.606 0.33057300
0.607 0.33066325
0.608 0.33075200
0.609 0.33083925
0.610 0.33092500
0.611 0.33100925
0.612 0.33109200
0.613 0.33117325
0.614 0.33125300
0.615 0.33133125
0.616 0.33140800
0.617 0.33148325
0.618 0.33155700
0.619 0.33162925
0.620 0.33170000
0.621 0.33176925
0.622 0.33183700
0.623 0.33190325
0.624 0.33196800
0.625 0.33203125
0.626 0.33209300
0.627 0.33215325
0.628 0.33221200
0.629 0.33226925
0.630 0.33232500
0.631 0.33237925
0.632 0.33243200
0.633 0.33248325
0.634 0.33253300
0.635 0.33258125
0.636 0.33262800
0.637 0.33267325
0.638 0.33271700
0.639 0.33275925
0.640 0.33280000
0.641 0.33283925
0.642 0.33287700
0.643 0.33291325
0.644 0.33294800
0.645 0.33298125
0.646 0.33301300
0.647 0.33304325
0.648 0.33307200
0.649 0.33309925
0.650 0.33312500
0.651 0.33314925
0.652 0.33317200
0.653 0.33319325
0.654 0.33321300
0.655 0.33323125
0.656 0.33324800
0.657 0.33326325
0.658 0.33327700
0.659 0.33328925
0.660 0.33330000
0.661 0.33330925
0.662 0.33331700
0.663 0.33332325
0.664 0.33332800
0.665 0.33333125
0.666 0.33333300
0.667 0.33333325
0.668 0.33333200
0.669 0.33332925
0.670 0.33332500
0.671 0.33331925
0.672 0.33331200
0.673 0.33330325
0.674 0.33329300
0.675 0.33328125
0.676 0.33326800
0.677 0.33325325
0.678 0.33323700
0.679 0.33321925
0.680 0.33320000
0.681 0.33317925
0.682 0.33315700
0.683 0.33313325
0.684 0.33310800
0.685 0.33308125
0.686 0.33305300
0.687 0.33302325
0.688 0.33299200
0.689 0.33295925
0.690 0.33292500
0.691 0.33288925
0.692 0.33285200
0.693 0.33281325
0.694 0.33277300
0.695 0.33273125
0.696 0.33268800
0.697 0.33264325
0.698 0.33259700
0.699 0.33254925
0.700 0.33250000
0.701 0.33244925
0.702 0.33239700
0.703 0.33234325
0.704 0.33228800
0.705 0.33223125
0.706 0.33217300
0.707 0.33211325
0.708 0.33205200
0.709 0.33198925
0.710 0.33192500
0.711 0.33185925
0.712 0.33179200
0.713 0.33172325
0.714 0.33165300
0.715 0.33158125
0.716 0.33150800
0.717 0.33143325
0.718 0.33135700
0.719 0.33127925
0.720 0.33120000
0.721 0.33111925
0.722 0.33103700
0.723 0.33095325
0.724 0.33086800
0.725 0.33078125
0.726 0.33069300
0.727 0.33060325
0.728 0.33051200
0.729 0.33041925
0.730 0.33032500
0.731 0.33022925
0.732 0.33013200
0.733 0.33003325
0.734 0.32993300
0.735 0.32983125
0.736 0.32972800
0.737 0.32962325
0.738 0.32951700
0.739 0.32940925
0.740 0.32930000
0.741 0.32918925
0.742 0.32907700
0.743 0.32896325
0.744 0.32884800
0.745 0.32873125
0.746 0.32861300
0.747 0.32849325
0.748 0.32837200
0.749 0.32824925
0.750 0.32812500
0.751 0.32799925
0.752 0.32787200
0.753 0.32774325
0.754 0.32761300
0.755 0.32748125
0.756 0.32734800
0.757 0.32721325
0.758 0.32707700
0.759 0.32693925
0.760 0.32680000
0.761 0.32665925
0.762 0.32651700
0.763 0.32637325
0.764 0.32622800
0.765 0.32608125
0.766 0.32593300
0.767 0.32578325
0.768 0.32563200
0.769 0.32547925
0.770 0.32532500
0.771 0.32516925
0.772 0.32501200
0.773 0.32485325
0.774 0.32469300
0.775 0.32453125
0.776 0.32436800
0.777 0.32420325
0.778 0.32403700
0.779 0.32386925
0.780 0.32370000
0.781 0.32352925
0.782 0.32335700
0.783 0.32318325
0.784 0.32300800
0.785 0.32283125
0.786 0.32265300
0.787 0.32247325
0.788 0.32229200
0.789 0.32210925
0.790 0.32192500
0.791 0.32173925
0.792 0.32155200
0.793 0.32136325
0.794 0.32117300
0.795 0.32098125
0.796 0.32078800
0.797 0.32059325
0.798 0.32039700
0.799 0.32019925
0.800 0.32000000
0.801 0.31979925
0.802 0.31959700
0.803 0.31939325
0.804 0.31918800
0.805 0.31898125
0.806 0.31877300
0.807 0.31856325
0.808 0.31835200
0.809 0.31813925
0.810 0.31792500
0.811 0.31770925
0.812 0.31749200
0.813 0.31727325
0.814 0.31705300
0.815 0.31683125
0.816 0.31660800
0.817 0.31638325
0.818 0.31615700
0.819 0.31592925
0.820 0.31570000
0.821 0.31546925
0.822 0.31523700
0.823 0.31500325
0.824 0.31476800
0.825 0.31453125
0.826 0.31429300
0.827 0.31405325
0.828 0.31381200
0.829 0.31356925
0.830 0.31332500
0.831 0.31307925
0.832 0.31283200
0.833 0.31258325
0.834 0.31233300
0.835 0.31208125
0.836 0.31182800
0.837 0.31157325
0.838 0.31131700
0.839 0.31105925
0.840 0.31080000
0.841 0.31053925
0.842 0.31027700
0.843 0.31001325
0.844 0.30974800
0.845 0.30948125
0.846 0.30921300
0.847 0.30894325
0.848 0.30867200
0.849 0.30839925
0.850 0.30812500
0.851 0.30784925
0.852 0.30757200
0.853 0.30729325
0.854 0.30701300
0.855 0.30673125
0.856 0.30644800
0.857 0.30616325
0.858 0.30587700
0.859 0.30558925
0.860 0.30530000
0.861 0.30500925
0.862 0.30471700
0.863 0.30442325
0.864 0.30412800
0.865 0.30383125
0.866 0.30353300
0.867 0.30323325
0.868 0.30293200
0.869 0.30262925
0.870 0.30232500
0.871 0.30201925
0.872 0.30171200
0.873 0.30140325
0.874 0.30109300
0.875 0.30078125
0.876 0.30046800
0.877 0.30015325
0.878 0.29983700
0.879 0.29951925
0.880 0.29920000
0.881 0.29887925
0.882 0.29855700
0.883 0.29823325
0.884 0.29790800
0.885 0.29758125
0.886 0.29725300
0.887 0.29692325
0.888 0.29659200
0.889 0.29625925
0.890 0.29592500
0.891 0.29558925
0.892 0.29525200
0.893 0.29491325
0.894 0.29457300
0.895 0.29423125
0.896 0.29388800
0.897 0.29354325
0.898 0.29319700
0.899 0.29284925
0.900 0.29250000
clc
for p=.1:.001:.9
p0=(1-p)^2;
p1=2*p*(1-p);
p2=p^2;
pWin=p1*.5+p2/4;
fprintf('%5.3f %10.8f\n',p,pWin)
end
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Posted by Charlie
on 2025-01-03 08:53:23 |
solution
