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 6 times and lost 4 times.

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