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 2 times and lost 8 times.

  1. Goat
  2. Goat
  3. Car
  4. Goat
  5. Goat
  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 350 times and lost 650 times.

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