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. Car
4. Car
5. Goat
6. Car
7. Car
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 336 times and lost 664 times.

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