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. Car
2. Goat
3. Car
4. Goat
5. Car
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 338 times and lost 662 times.

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