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 5 times and lost 5 times.

1. Car
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Car
8. Goat
9. Goat
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 338 times and lost 662 times.

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