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. Goat
  6. Car
  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 322 times and lost 678 times.

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