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. Goat
2. Car
3. Goat
4. Goat
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 337 times and lost 663 times.

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