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. Car
  4. Goat
  5. Car
  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 294 times and lost 706 times.

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