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. Car
5. Car
6. Car
7. Goat
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 326 times and lost 674 times.

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