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 6 times and lost 4 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Car
7. Car
8. Car
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 339 times and lost 661 times.

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