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. Car
  2. Goat
  3. Goat
  4. Car
  5. Goat
  6. Goat
  7. Goat
  8. Goat
  9. Goat
  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 338 times and lost 662 times.

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