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 3 times and lost 7 times.

  1. Goat
  2. Goat
  3. Car
  4. Goat
  5. Goat
  6. Goat
  7. Goat
  8. Goat
  9. Car
  10. Car

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 343 times and lost 657 times.

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