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. Goat
3. Car
4. Car
5. Goat
6. Car
7. Car
8. Goat
9. Goat
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 342 times and lost 658 times.

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