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. Car
  2. Goat
  3. Car
  4. Goat
  5. Goat
  6. Car
  7. Car
  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 342 times and lost 658 times.

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