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 5 times and lost 5 times.

1. Goat
2. Car
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Goat
9. Car
10. Car

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 318 times and lost 682 times.

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