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 5 times and lost 5 times.

  1. Goat
  2. Car
  3. Goat
  4. Goat
  5. Car
  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 355 times and lost 645 times.

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