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. Goat
  3. Goat
  4. Car
  5. Goat
  6. Goat
  7. Car
  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 361 times and lost 639 times.

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