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. Car
2. Car
3. Goat
4. Goat
5. Car
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 342 times and lost 658 times.

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