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. Goat
  2. Goat
  3. Car
  4. Goat
  5. Goat
  6. Goat
  7. Car
  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 351 times and lost 649 times.

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