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. Car
  2. Goat
  3. Car
  4. Goat
  5. Goat
  6. Goat
  7. Goat
  8. Car
  9. Car
  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 347 times and lost 653 times.

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