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 4 times and lost 6 times.

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

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 346 times and lost 654 times.

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

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 112 times and lost 888 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 38 times and lost 62 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;
	}
}