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. Car
7. Goat
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 331 times and lost 669 times.

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