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 5 times and lost 5 times.

  1. Goat
  2. Goat
  3. Car
  4. Car
  5. Goat
  6. Goat
  7. Car
  8. Car
  9. Goat
  10. Car

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 346 times and lost 654 times.

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