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 8 times and lost 2 times.

  1. Car
  2. Car
  3. Car
  4. Car
  5. Car
  6. Goat
  7. Goat
  8. Car
  9. Car
  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 341 times and lost 659 times.

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