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 2 times and lost 8 times.

  1. Goat
  2. Car
  3. Goat
  4. Goat
  5. Goat
  6. Goat
  7. Goat
  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 370 times and lost 630 times.

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