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. Car
  2. Goat
  3. Car
  4. Car
  5. Goat
  6. Goat
  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 363 times and lost 637 times.

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