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. Goat
3. Goat
4. Goat
5. Car
6. Car
7. Car
8. Goat
9. Car
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 335 times and lost 665 times.

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