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 5 times and lost 5 times.

  1. Goat
  2. Car
  3. Goat
  4. Car
  5. Car
  6. Car
  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 359 times and lost 641 times.

  1. Car
  2. Goat
  3. Goat
  4. Goat
  5. Goat
  6. Goat
  7. Goat
  8. Goat
  9. Goat
  10. Goat
  11. Goat
  12. Goat
  13. Goat
  14. Car
  15. Car
  16. Goat
  17. Goat
  18. Goat
  19. Goat
  20. Goat
  21. Goat
  22. Car
  23. Goat
  24. Car
  25. Goat
  26. Goat
  27. Car
  28. Goat
  29. Goat
  30. Goat
  31. Goat
  32. Goat
  33. Car
  34. Car
  35. Goat
  36. Goat
  37. Goat
  38. Goat
  39. Goat
  40. Goat
  41. Goat
  42. Goat
  43. Goat
  44. Car
  45. Goat
  46. Car
  47. Car
  48. Car
  49. Goat
  50. Car
  51. Car
  52. Car
  53. Car
  54. Car
  55. Car
  56. Goat
  57. Goat
  58. Goat
  59. Car
  60. Goat
  61. Goat
  62. Car
  63. Car
  64. Car
  65. Goat
  66. Goat
  67. Car
  68. Goat
  69. Car
  70. Goat
  71. Car
  72. Goat
  73. Goat
  74. Car
  75. Goat
  76. Goat
  77. Goat
  78. Car
  79. Car
  80. Goat
  81. Goat
  82. Car
  83. Car
  84. Goat
  85. Car
  86. Goat
  87. Goat
  88. Goat
  89. Goat
  90. Goat
  91. Goat
  92. Goat
  93. Goat
  94. Goat
  95. Car
  96. Goat
  97. Goat
  98. Goat
  99. Goat
  100. Goat
  101. Goat
  102. Goat
  103. Goat
  104. Car
  105. Goat
  106. Car
  107. Goat
  108. Goat
  109. Goat
  110. Goat
  111. Goat
  112. Goat
  113. Goat
  114. Car
  115. Goat
  116. Car
  117. Goat
  118. Goat
  119. Car
  120. Goat
  121. Goat
  122. Car
  123. Car
  124. Goat
  125. Goat
  126. Car
  127. Car
  128. Car
  129. Goat
  130. Goat
  131. Car
  132. Goat
  133. Goat
  134. Goat
  135. Goat
  136. Goat
  137. Goat
  138. Goat
  139. Goat
  140. Goat
  141. Car
  142. Goat
  143. Car
  144. Car
  145. Goat
  146. Goat
  147. Goat
  148. Goat
  149. Car
  150. Car
  151. Goat
  152. Goat
  153. Car
  154. Goat
  155. Goat
  156. Car
  157. Goat
  158. Car
  159. Goat
  160. Goat
  161. Car
  162. Goat
  163. Goat
  164. Car
  165. Goat
  166. Goat
  167. Goat
  168. Goat
  169. Car
  170. Car
  171. Goat
  172. Car
  173. Goat
  174. Car
  175. Goat
  176. Car
  177. Goat
  178. Goat
  179. Goat
  180. Car
  181. Goat
  182. Car
  183. Goat
  184. Goat
  185. Car
  186. Goat
  187. Goat
  188. Goat
  189. Goat
  190. Car
  191. Car
  192. Goat
  193. Goat
  194. Goat
  195. Car
  196. Goat
  197. Goat
  198. Goat
  199. Goat
  200. Car
  201. Goat
  202. Goat
  203. Goat
  204. Car
  205. Goat
  206. Goat
  207. Car
  208. Goat
  209. Goat
  210. Car
  211. Car
  212. Car
  213. Goat
  214. Goat
  215. Goat
  216. Goat
  217. Goat
  218. Car
  219. Car
  220. Goat
  221. Goat
  222. Car
  223. Goat
  224. Goat
  225. Goat
  226. Goat
  227. Car
  228. Goat
  229. Goat
  230. Car
  231. Car
  232. Car
  233. Car
  234. Car
  235. Goat
  236. Car
  237. Goat
  238. Car
  239. Goat
  240. Car
  241. Car
  242. Car
  243. Goat
  244. Car
  245. Car
  246. Goat
  247. Goat
  248. Goat
  249. Goat
  250. Car
  251. Car
  252. Car
  253. Goat
  254. Car
  255. Goat
  256. Car
  257. Goat
  258. Goat
  259. Goat
  260. Car
  261. Car
  262. Goat
  263. Car
  264. Car
  265. Goat
  266. Car
  267. Goat
  268. Car
  269. Goat
  270. Goat
  271. Goat
  272. Goat
  273. Goat
  274. Goat
  275. Goat
  276. Goat
  277. Goat
  278. Goat
  279. Goat
  280. Goat
  281. Goat
  282. Goat
  283. Car
  284. Goat
  285. Car
  286. Goat
  287. Goat
  288. Car
  289. Car
  290. Car
  291. Car
  292. Car
  293. Goat
  294. Goat
  295. Car
  296. Car
  297. Goat
  298. Car
  299. Car
  300. Goat
  301. Car
  302. Goat
  303. Car
  304. Goat
  305. Goat
  306. Goat
  307. Goat
  308. Car
  309. Car
  310. Goat
  311. Car
  312. Car
  313. Car
  314. Goat
  315. Car
  316. Car
  317. Goat
  318. Goat
  319. Goat
  320. Car
  321. Car
  322. Goat
  323. Goat
  324. Goat
  325. Car
  326. Car
  327. Car
  328. Goat
  329. Goat
  330. Goat
  331. Goat
  332. Car
  333. Goat
  334. Car
  335. Goat
  336. Car
  337. Goat
  338. Goat
  339. Goat
  340. Car
  341. Car
  342. Car
  343. Goat
  344. Goat
  345. Goat
  346. Goat
  347. Car
  348. Goat
  349. Car
  350. Goat
  351. Goat
  352. Goat
  353. Car
  354. Goat
  355. Car
  356. Car
  357. Goat
  358. Car
  359. Goat
  360. Car
  361. Goat
  362. Goat
  363. Goat
  364. Goat
  365. Goat
  366. Goat
  367. Goat
  368. Goat
  369. Goat
  370. Car
  371. Goat
  372. Car
  373. Goat
  374. Car
  375. Goat
  376. Goat
  377. Car
  378. Goat
  379. Goat
  380. Goat
  381. Car
  382. Goat
  383. Goat
  384. Goat
  385. Goat
  386. Goat
  387. Goat
  388. Goat
  389. Goat
  390. Goat
  391. Goat
  392. Goat
  393. Goat
  394. Car
  395. Car
  396. Goat
  397. Car
  398. Car
  399. Goat
  400. Goat
  401. Goat
  402. Goat
  403. Goat
  404. Goat
  405. Car
  406. Goat
  407. Goat
  408. Car
  409. Car
  410. Goat
  411. Goat
  412. Goat
  413. Car
  414. Goat
  415. Car
  416. Goat
  417. Goat
  418. Goat
  419. Goat
  420. Car
  421. Goat
  422. Goat
  423. Goat
  424. Goat
  425. Car
  426. Goat
  427. Goat
  428. Goat
  429. Goat
  430. Goat
  431. Goat
  432. Goat
  433. Goat
  434. Car
  435. Goat
  436. Goat
  437. Car
  438. Car
  439. Goat
  440. Car
  441. Goat
  442. Goat
  443. Car
  444. Car
  445. Car
  446. Goat
  447. Goat
  448. Goat
  449. Goat
  450. Goat
  451. Goat
  452. Goat
  453. Goat
  454. Car
  455. Car
  456. Car
  457. Car
  458. Goat
  459. Car
  460. Goat
  461. Goat
  462. Car
  463. Goat
  464. Goat
  465. Car
  466. Goat
  467. Car
  468. Goat
  469. Car
  470. Goat
  471. Goat
  472. Goat
  473. Goat
  474. Goat
  475. Goat
  476. Car
  477. Goat
  478. Car
  479. Car
  480. Car
  481. Car
  482. Goat
  483. Goat
  484. Car
  485. Car
  486. Goat
  487. Goat
  488. Car
  489. Goat
  490. Goat
  491. Goat
  492. Goat
  493. Car
  494. Goat
  495. Car
  496. Goat
  497. Car
  498. Car
  499. Goat
  500. Goat
  501. Goat
  502. Goat
  503. Car
  504. Goat
  505. Car
  506. Car
  507. Car
  508. Car
  509. Goat
  510. Car
  511. Goat
  512. Car
  513. Goat
  514. Goat
  515. Goat
  516. Car
  517. Car
  518. Car
  519. Goat
  520. Goat
  521. Car
  522. Car
  523. Car
  524. Car
  525. Goat
  526. Car
  527. Goat
  528. Goat
  529. Car
  530. Goat
  531. Goat
  532. Goat
  533. Goat
  534. Goat
  535. Goat
  536. Car
  537. Car
  538. Goat
  539. Car
  540. Car
  541. Goat
  542. Goat
  543. Car
  544. Goat
  545. Goat
  546. Goat
  547. Goat
  548. Goat
  549. Car
  550. Car
  551. Car
  552. Car
  553. Car
  554. Goat
  555. Car
  556. Goat
  557. Car
  558. Goat
  559. Car
  560. Car
  561. Goat
  562. Car
  563. Car
  564. Goat
  565. Car
  566. Goat
  567. Car
  568. Goat
  569. Goat
  570. Goat
  571. Car
  572. Goat
  573. Car
  574. Car
  575. Goat
  576. Goat
  577. Car
  578. Goat
  579. Goat
  580. Goat
  581. Goat
  582. Goat
  583. Goat
  584. Car
  585. Goat
  586. Goat
  587. Goat
  588. Goat
  589. Goat
  590. Goat
  591. Car
  592. Goat
  593. Car
  594. Goat
  595. Goat
  596. Goat
  597. Car
  598. Car
  599. Car
  600. Goat
  601. Goat
  602. Goat
  603. Goat
  604. Goat
  605. Car
  606. Goat
  607. Goat
  608. Goat
  609. Car
  610. Goat
  611. Car
  612. Goat
  613. Car
  614. Car
  615. Car
  616. Car
  617. Goat
  618. Car
  619. Car
  620. Goat
  621. Goat
  622. Car
  623. Goat
  624. Goat
  625. Goat
  626. Goat
  627. Car
  628. Goat
  629. Goat
  630. Goat
  631. Goat
  632. Goat
  633. Goat
  634. Car
  635. Car
  636. Goat
  637. Goat
  638. Goat
  639. Goat
  640. Goat
  641. Goat
  642. Car
  643. Goat
  644. Goat
  645. Car
  646. Car
  647. Goat
  648. Car
  649. Goat
  650. Car
  651. Goat
  652. Car
  653. Car
  654. Car
  655. Goat
  656. Goat
  657. Goat
  658. Goat
  659. Car
  660. Goat
  661. Goat
  662. Goat
  663. Car
  664. Goat
  665. Car
  666. Goat
  667. Car
  668. Goat
  669. Goat
  670. Goat
  671. Goat
  672. Car
  673. Car
  674. Goat
  675. Car
  676. Goat
  677. Goat
  678. Car
  679. Car
  680. Goat
  681. Car
  682. Goat
  683. Car
  684. Goat
  685. Car
  686. Car
  687. Goat
  688. Car
  689. Car
  690. Goat
  691. Car
  692. Car
  693. Goat
  694. Goat
  695. Goat
  696. Car
  697. Car
  698. Car
  699. Goat
  700. Car
  701. Goat
  702. Car
  703. Goat
  704. Goat
  705. Goat
  706. Goat
  707. Goat
  708. Car
  709. Goat
  710. Goat
  711. Goat
  712. Car
  713. Car
  714. Goat
  715. Goat
  716. Goat
  717. Goat
  718. Car
  719. Goat
  720. Car
  721. Goat
  722. Car
  723. Car
  724. Goat
  725. Goat
  726. Goat
  727. Goat
  728. Car
  729. Goat
  730. Goat
  731. Goat
  732. Car
  733. Goat
  734. Car
  735. Goat
  736. Goat
  737. Goat
  738. Goat
  739. Car
  740. Goat
  741. Goat
  742. Goat
  743. Car
  744. Goat
  745. Goat
  746. Goat
  747. Goat
  748. Goat
  749. Goat
  750. Goat
  751. Goat
  752. Goat
  753. Goat
  754. Car
  755. Car
  756. Goat
  757. Goat
  758. Goat
  759. Car
  760. Goat
  761. Goat
  762. Goat
  763. Goat
  764. Goat
  765. Goat
  766. Goat
  767. Car
  768. Car
  769. Goat
  770. Goat
  771. Goat
  772. Goat
  773. Goat
  774. Goat
  775. Goat
  776. Goat
  777. Goat
  778. Goat
  779. Goat
  780. Car
  781. Goat
  782. Goat
  783. Car
  784. Goat
  785. Goat
  786. Goat
  787. Car
  788. Goat
  789. Car
  790. Goat
  791. Goat
  792. Car
  793. Car
  794. Goat
  795. Goat
  796. Goat
  797. Car
  798. Goat
  799. Goat
  800. Goat
  801. Goat
  802. Goat
  803. Car
  804. Goat
  805. Goat
  806. Car
  807. Goat
  808. Goat
  809. Car
  810. Goat
  811. Goat
  812. Goat
  813. Goat
  814. Goat
  815. Goat
  816. Car
  817. Car
  818. Goat
  819. Goat
  820. Car
  821. Goat
  822. Car
  823. Goat
  824. Goat
  825. Goat
  826. Car
  827. Goat
  828. Car
  829. Goat
  830. Goat
  831. Car
  832. Goat
  833. Goat
  834. Goat
  835. Car
  836. Car
  837. Goat
  838. Car
  839. Goat
  840. Car
  841. Goat
  842. Goat
  843. Car
  844. Goat
  845. Goat
  846. Goat
  847. Goat
  848. Goat
  849. Goat
  850. Goat
  851. Goat
  852. Goat
  853. Car
  854. Goat
  855. Goat
  856. Goat
  857. Car
  858. Goat
  859. Goat
  860. Goat
  861. Car
  862. Goat
  863. Goat
  864. Goat
  865. Car
  866. Goat
  867. Car
  868. Goat
  869. Goat
  870. Goat
  871. Car
  872. Car
  873. Car
  874. Goat
  875. Car
  876. Goat
  877. Car
  878. Goat
  879. Goat
  880. Car
  881. Car
  882. Car
  883. Goat
  884. Car
  885. Car
  886. Goat
  887. Car
  888. Goat
  889. Goat
  890. Goat
  891. Car
  892. Goat
  893. Goat
  894. Car
  895. Goat
  896. Goat
  897. Car
  898. Car
  899. Goat
  900. Goat
  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. Goat
  914. Goat
  915. Car
  916. Car
  917. Goat
  918. Goat
  919. Car
  920. Car
  921. Goat
  922. Goat
  923. Car
  924. Goat
  925. Car
  926. Goat
  927. Goat
  928. Car
  929. Goat
  930. Car
  931. Goat
  932. Goat
  933. Goat
  934. Car
  935. Car
  936. Goat
  937. Car
  938. Goat
  939. Goat
  940. Goat
  941. Car
  942. Car
  943. Goat
  944. Car
  945. Goat
  946. Goat
  947. Car
  948. Goat
  949. Goat
  950. Goat
  951. Goat
  952. Goat
  953. Car
  954. Goat
  955. Car
  956. Car
  957. Goat
  958. Goat
  959. Goat
  960. Car
  961. Goat
  962. Car
  963. Goat
  964. Goat
  965. Goat
  966. Goat
  967. Goat
  968. Goat
  969. Goat
  970. Car
  971. Goat
  972. Goat
  973. Car
  974. Car
  975. Goat
  976. Goat
  977. Car
  978. Goat
  979. Goat
  980. Goat
  981. Goat
  982. Goat
  983. Goat
  984. Car
  985. Car
  986. Car
  987. Goat
  988. Goat
  989. Car
  990. Car
  991. Goat
  992. Goat
  993. Goat
  994. Goat
  995. Goat
  996. Car
  997. Car
  998. Goat
  999. Goat
  1000. Car

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 83 times and lost 917 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;
	}
}