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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Goat
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 327 times and lost 673 times.

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