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 3 times and lost 7 times.

  1. Goat
  2. Car
  3. Goat
  4. Car
  5. Goat
  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 349 times and lost 651 times.

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