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. Car
6. Goat
7. Car
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 364 times and lost 636 times.

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

Custom test result

Player won 35 times and lost 65 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;
	}
}