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 1 time and lost 9 times.

  1. Goat
  2. Goat
  3. Goat
  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 332 times and lost 668 times.

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