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. Car
  6. Goat
  7. Goat
  8. Car
  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 323 times and lost 677 times.

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