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 4 times and lost 6 times.

  1. Goat
  2. Car
  3. Goat
  4. Car
  5. Goat
  6. Car
  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 345 times and lost 655 times.

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