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. Car
  7. Car
  8. Car
  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 349 times and lost 651 times.

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