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