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. Goat
  5. Car
  6. Goat
  7. Goat
  8. Car
  9. Car
  10. Car

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