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. Goat
  4. Goat
  5. Goat
  6. Car
  7. Goat
  8. Goat
  9. Car
  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 348 times and lost 652 times.

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