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

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