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. Goat
2. Goat
3. Car
4. Car
5. Goat
6. Car
7. Goat
8. Car
9. Car
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 336 times and lost 664 times.

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