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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
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 325 times and lost 675 times.

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