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. Goat
  3. Goat
  4. Car
  5. Goat
  6. Car
  7. Goat
  8. Car
  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 353 times and lost 647 times.

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