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 2 times and lost 8 times.

  1. Car
  2. Car
  3. Goat
  4. Goat
  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 346 times and lost 654 times.

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