Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

  1. Car
  2. Goat
  3. Goat
  4. Goat
  5. Car
  6. Goat
  7. Goat
  8. Goat
  9. Goat
  10. Car

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 326 times and lost 674 times.

  1. Goat
  2. Car
  3. Goat
  4. Car
  5. Goat
  6. Goat
  7. Car
  8. Goat
  9. Car
  10. Car
  11. Car
  12. Goat
  13. Goat
  14. Goat
  15. Goat
  16. Goat
  17. Goat
  18. Goat
  19. Goat
  20. Goat
  21. Car
  22. Goat
  23. Goat
  24. Goat
  25. Goat
  26. Car
  27. Goat
  28. Goat
  29. Goat
  30. Car
  31. Goat
  32. Goat
  33. Car
  34. Goat
  35. Goat
  36. Goat
  37. Car
  38. Goat
  39. Goat
  40. Goat
  41. Goat
  42. Goat
  43. Goat
  44. Goat
  45. Car
  46. Goat
  47. Goat
  48. Car
  49. Car
  50. Goat
  51. Goat
  52. Car
  53. Goat
  54. Goat
  55. Goat
  56. Car
  57. Goat
  58. Goat
  59. Car
  60. Goat
  61. Goat
  62. Car
  63. Goat
  64. Goat
  65. Goat
  66. Goat
  67. Goat
  68. Goat
  69. Goat
  70. Goat
  71. Goat
  72. Goat
  73. Goat
  74. Car
  75. Car
  76. Goat
  77. Car
  78. Car
  79. Car
  80. Car
  81. Goat
  82. Goat
  83. Goat
  84. Goat
  85. Car
  86. Car
  87. Goat
  88. Goat
  89. Goat
  90. Car
  91. Car
  92. Goat
  93. Goat
  94. Goat
  95. Goat
  96. Goat
  97. Car
  98. Car
  99. Car
  100. Goat
  101. Car
  102. Car
  103. Goat
  104. Goat
  105. Goat
  106. Goat
  107. Goat
  108. Car
  109. Goat
  110. Goat
  111. Car
  112. Goat
  113. Goat
  114. Car
  115. Car
  116. Goat
  117. Goat
  118. Goat
  119. Car
  120. Goat
  121. Goat
  122. Goat
  123. Goat
  124. Goat
  125. Car
  126. Goat
  127. Goat
  128. Goat
  129. Goat
  130. Goat
  131. Goat
  132. Car
  133. Car
  134. Goat
  135. Car
  136. Goat
  137. Car
  138. Goat
  139. Car
  140. Goat
  141. Goat
  142. Car
  143. Goat
  144. Goat
  145. Goat
  146. Goat
  147. Car
  148. Car
  149. Car
  150. Goat
  151. Goat
  152. Goat
  153. Car
  154. Goat
  155. Goat
  156. Car
  157. Goat
  158. Goat
  159. Goat
  160. Goat
  161. Goat
  162. Goat
  163. Goat
  164. Goat
  165. Goat
  166. Goat
  167. Goat
  168. Goat
  169. Car
  170. Goat
  171. Goat
  172. Goat
  173. Car
  174. Goat
  175. Car
  176. Goat
  177. Goat
  178. Goat
  179. Goat
  180. Goat
  181. Goat
  182. Goat
  183. Goat
  184. Car
  185. Goat
  186. Goat
  187. Car
  188. Goat
  189. Car
  190. Goat
  191. Goat
  192. Goat
  193. Car
  194. Car
  195. Goat
  196. Goat
  197. Goat
  198. Goat
  199. Goat
  200. Car
  201. Car
  202. Goat
  203. Car
  204. Goat
  205. Goat
  206. Goat
  207. Goat
  208. Car
  209. Goat
  210. Car
  211. Goat
  212. Goat
  213. Goat
  214. Car
  215. Goat
  216. Goat
  217. Goat
  218. Goat
  219. Goat
  220. Goat
  221. Goat
  222. Car
  223. Goat
  224. Car
  225. Goat
  226. Goat
  227. Car
  228. Car
  229. Goat
  230. Car
  231. Goat
  232. Goat
  233. Goat
  234. Goat
  235. Goat
  236. Car
  237. Goat
  238. Car
  239. Goat
  240. Goat
  241. Goat
  242. Goat
  243. Goat
  244. Goat
  245. Car
  246. Goat
  247. Goat
  248. Goat
  249. Goat
  250. Goat
  251. Car
  252. Car
  253. Goat
  254. Car
  255. Goat
  256. Car
  257. Goat
  258. Car
  259. Goat
  260. Goat
  261. Goat
  262. Goat
  263. Goat
  264. Goat
  265. Goat
  266. Goat
  267. Car
  268. Car
  269. Goat
  270. Goat
  271. Goat
  272. Car
  273. Goat
  274. Goat
  275. Goat
  276. Car
  277. Car
  278. Goat
  279. Car
  280. Goat
  281. Goat
  282. Car
  283. Goat
  284. Goat
  285. Car
  286. Goat
  287. Goat
  288. Goat
  289. Goat
  290. Goat
  291. Car
  292. Goat
  293. Goat
  294. Goat
  295. Goat
  296. Goat
  297. Goat
  298. Car
  299. Goat
  300. Goat
  301. Car
  302. Goat
  303. Goat
  304. Car
  305. Car
  306. Goat
  307. Goat
  308. Goat
  309. Car
  310. Goat
  311. Goat
  312. Goat
  313. Car
  314. Car
  315. Goat
  316. Goat
  317. Goat
  318. Goat
  319. Goat
  320. Goat
  321. Goat
  322. Car
  323. Goat
  324. Goat
  325. Goat
  326. Goat
  327. Goat
  328. Goat
  329. Car
  330. Goat
  331. Car
  332. Goat
  333. Goat
  334. Car
  335. Car
  336. Car
  337. Car
  338. Car
  339. Car
  340. Goat
  341. Car
  342. Goat
  343. Car
  344. Car
  345. Car
  346. Car
  347. Goat
  348. Goat
  349. Goat
  350. Car
  351. Car
  352. Goat
  353. Goat
  354. Goat
  355. Goat
  356. Car
  357. Goat
  358. Car
  359. Car
  360. Goat
  361. Goat
  362. Car
  363. Goat
  364. Car
  365. Goat
  366. Goat
  367. Goat
  368. Goat
  369. Goat
  370. Goat
  371. Goat
  372. Car
  373. Goat
  374. Goat
  375. Car
  376. Car
  377. Car
  378. Car
  379. Goat
  380. Car
  381. Car
  382. Car
  383. Goat
  384. Goat
  385. Car
  386. Goat
  387. Car
  388. Goat
  389. Goat
  390. Car
  391. Car
  392. Goat
  393. Car
  394. Goat
  395. Goat
  396. Goat
  397. Goat
  398. Goat
  399. Goat
  400. Goat
  401. Car
  402. Goat
  403. Goat
  404. Goat
  405. Goat
  406. Goat
  407. Car
  408. Goat
  409. Goat
  410. Goat
  411. Car
  412. Car
  413. Car
  414. Goat
  415. Goat
  416. Car
  417. Car
  418. Goat
  419. Car
  420. Car
  421. Car
  422. Goat
  423. Goat
  424. Car
  425. Goat
  426. Car
  427. Goat
  428. Goat
  429. Goat
  430. Car
  431. Goat
  432. Car
  433. Car
  434. Goat
  435. Goat
  436. Goat
  437. Goat
  438. Car
  439. Goat
  440. Goat
  441. Car
  442. Car
  443. Goat
  444. Goat
  445. Goat
  446. Car
  447. Car
  448. Goat
  449. Car
  450. Goat
  451. Goat
  452. Car
  453. Goat
  454. Goat
  455. Goat
  456. Goat
  457. Goat
  458. Goat
  459. Car
  460. Goat
  461. Goat
  462. Goat
  463. Goat
  464. Car
  465. Goat
  466. Goat
  467. Goat
  468. Goat
  469. Goat
  470. Goat
  471. Car
  472. Car
  473. Goat
  474. Car
  475. Goat
  476. Goat
  477. Goat
  478. Car
  479. Goat
  480. Goat
  481. Goat
  482. Goat
  483. Car
  484. Goat
  485. Goat
  486. Goat
  487. Goat
  488. Goat
  489. Car
  490. Car
  491. Car
  492. Car
  493. Goat
  494. Goat
  495. Car
  496. Car
  497. Car
  498. Goat
  499. Goat
  500. Goat
  501. Goat
  502. Goat
  503. Car
  504. Goat
  505. Goat
  506. Car
  507. Goat
  508. Goat
  509. Car
  510. Goat
  511. Car
  512. Car
  513. Car
  514. Goat
  515. Goat
  516. Car
  517. Goat
  518. Goat
  519. Goat
  520. Goat
  521. Goat
  522. Goat
  523. Goat
  524. Car
  525. Goat
  526. Goat
  527. Goat
  528. Goat
  529. Car
  530. Goat
  531. Goat
  532. Car
  533. Car
  534. Goat
  535. Goat
  536. Goat
  537. Goat
  538. Car
  539. Goat
  540. Goat
  541. Car
  542. Car
  543. Goat
  544. Car
  545. Goat
  546. Goat
  547. Car
  548. Goat
  549. Goat
  550. Car
  551. Goat
  552. Goat
  553. Goat
  554. Goat
  555. Goat
  556. Goat
  557. Goat
  558. Goat
  559. Goat
  560. Car
  561. Goat
  562. Goat
  563. Goat
  564. Goat
  565. Goat
  566. Goat
  567. Goat
  568. Goat
  569. Goat
  570. Goat
  571. Car
  572. Car
  573. Goat
  574. Goat
  575. Goat
  576. Goat
  577. Goat
  578. Car
  579. Goat
  580. Goat
  581. Car
  582. Car
  583. Car
  584. Car
  585. Goat
  586. Car
  587. Goat
  588. Car
  589. Car
  590. Goat
  591. Car
  592. Goat
  593. Goat
  594. Goat
  595. Goat
  596. Goat
  597. Car
  598. Car
  599. Car
  600. Goat
  601. Goat
  602. Car
  603. Goat
  604. Goat
  605. Goat
  606. Car
  607. Car
  608. Goat
  609. Goat
  610. Car
  611. Goat
  612. Car
  613. Goat
  614. Car
  615. Car
  616. Goat
  617. Goat
  618. Goat
  619. Car
  620. Car
  621. Goat
  622. Car
  623. Goat
  624. Goat
  625. Car
  626. Goat
  627. Goat
  628. Goat
  629. Car
  630. Goat
  631. Car
  632. Goat
  633. Goat
  634. Car
  635. Goat
  636. Goat
  637. Car
  638. Car
  639. Goat
  640. Goat
  641. Goat
  642. Goat
  643. Goat
  644. Goat
  645. Car
  646. Goat
  647. Goat
  648. Car
  649. Goat
  650. Goat
  651. Goat
  652. Goat
  653. Goat
  654. Car
  655. Goat
  656. Car
  657. Car
  658. Goat
  659. Goat
  660. Goat
  661. Goat
  662. Goat
  663. Car
  664. Goat
  665. Car
  666. Goat
  667. Goat
  668. Car
  669. Goat
  670. Goat
  671. Car
  672. Goat
  673. Car
  674. Car
  675. Goat
  676. Goat
  677. Goat
  678. Goat
  679. Goat
  680. Car
  681. Goat
  682. Goat
  683. Car
  684. Car
  685. Goat
  686. Car
  687. Goat
  688. Goat
  689. Goat
  690. Goat
  691. Car
  692. Car
  693. Goat
  694. Car
  695. Car
  696. Car
  697. Goat
  698. Goat
  699. Goat
  700. Goat
  701. Goat
  702. Goat
  703. Goat
  704. Goat
  705. Goat
  706. Car
  707. Goat
  708. Goat
  709. Goat
  710. Goat
  711. Car
  712. Car
  713. Goat
  714. Goat
  715. Goat
  716. Car
  717. Goat
  718. Goat
  719. Goat
  720. Car
  721. Car
  722. Car
  723. Goat
  724. Goat
  725. Goat
  726. Goat
  727. Goat
  728. Goat
  729. Goat
  730. Goat
  731. Goat
  732. Goat
  733. Goat
  734. Car
  735. Goat
  736. Goat
  737. Goat
  738. Goat
  739. Goat
  740. Goat
  741. Car
  742. Car
  743. Car
  744. Car
  745. Car
  746. Goat
  747. Car
  748. Car
  749. Goat
  750. Car
  751. Car
  752. Goat
  753. Goat
  754. Goat
  755. Goat
  756. Goat
  757. Car
  758. Car
  759. Car
  760. Goat
  761. Goat
  762. Goat
  763. Goat
  764. Goat
  765. Car
  766. Goat
  767. Goat
  768. Goat
  769. Goat
  770. Goat
  771. Goat
  772. Goat
  773. Goat
  774. Goat
  775. Goat
  776. Goat
  777. Car
  778. Car
  779. Car
  780. Goat
  781. Goat
  782. Car
  783. Goat
  784. Goat
  785. Goat
  786. Goat
  787. Goat
  788. Goat
  789. Car
  790. Goat
  791. Goat
  792. Goat
  793. Car
  794. Goat
  795. Goat
  796. Goat
  797. Goat
  798. Goat
  799. Goat
  800. Goat
  801. Goat
  802. Car
  803. Goat
  804. Car
  805. Car
  806. Goat
  807. Goat
  808. Goat
  809. Goat
  810. Car
  811. Goat
  812. Goat
  813. Car
  814. Car
  815. Goat
  816. Goat
  817. Goat
  818. Goat
  819. Goat
  820. Goat
  821. Goat
  822. Car
  823. Car
  824. Goat
  825. Goat
  826. Goat
  827. Car
  828. Car
  829. Goat
  830. Car
  831. Goat
  832. Goat
  833. Goat
  834. Car
  835. Goat
  836. Goat
  837. Goat
  838. Car
  839. Goat
  840. Goat
  841. Goat
  842. Goat
  843. Goat
  844. Goat
  845. Car
  846. Goat
  847. Goat
  848. Goat
  849. Goat
  850. Goat
  851. Car
  852. Car
  853. Goat
  854. Goat
  855. Car
  856. Car
  857. Car
  858. Goat
  859. Goat
  860. Car
  861. Goat
  862. Goat
  863. Goat
  864. Goat
  865. Car
  866. Goat
  867. Goat
  868. Goat
  869. Car
  870. Goat
  871. Car
  872. Goat
  873. Goat
  874. Goat
  875. Car
  876. Goat
  877. Car
  878. Goat
  879. Car
  880. Car
  881. Goat
  882. Car
  883. Goat
  884. Car
  885. Car
  886. Car
  887. Goat
  888. Goat
  889. Goat
  890. Goat
  891. Goat
  892. Goat
  893. Goat
  894. Goat
  895. Goat
  896. Goat
  897. Goat
  898. Goat
  899. Goat
  900. Car
  901. Car
  902. Car
  903. Car
  904. Goat
  905. Car
  906. Goat
  907. Goat
  908. Goat
  909. Goat
  910. Goat
  911. Goat
  912. Car
  913. Goat
  914. Goat
  915. Goat
  916. Goat
  917. Goat
  918. Goat
  919. Car
  920. Goat
  921. Car
  922. Goat
  923. Goat
  924. Goat
  925. Goat
  926. Goat
  927. Goat
  928. Goat
  929. Goat
  930. Car
  931. Goat
  932. Car
  933. Car
  934. Goat
  935. Goat
  936. Goat
  937. Goat
  938. Goat
  939. Goat
  940. Goat
  941. Goat
  942. Goat
  943. Car
  944. Goat
  945. Goat
  946. Car
  947. Car
  948. Car
  949. Car
  950. Goat
  951. Goat
  952. Car
  953. Car
  954. Car
  955. Goat
  956. Car
  957. Car
  958. Goat
  959. Goat
  960. Goat
  961. Goat
  962. Goat
  963. Goat
  964. Car
  965. Goat
  966. Goat
  967. Car
  968. Car
  969. Car
  970. Car
  971. Car
  972. Goat
  973. Goat
  974. Car
  975. Goat
  976. Goat
  977. Car
  978. Car
  979. Goat
  980. Car
  981. Car
  982. Car
  983. Goat
  984. Goat
  985. Goat
  986. Car
  987. Car
  988. Goat
  989. Goat
  990. Goat
  991. Goat
  992. Goat
  993. Goat
  994. Car
  995. Goat
  996. Goat
  997. Car
  998. Goat
  999. Goat
  1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 106 times and lost 894 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

How many doors?

How many iterations?

Should the player stay with their original choice?
Yes     No

Show images?
Yes     No

Custom test result

Player won 34 times and lost 66 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}