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. Goat
  5. Car
  6. Goat
  7. Car
  8. Goat
  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 366 times and lost 634 times.

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