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 4 times and lost 6 times.

1. Car
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
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 330 times and lost 670 times.

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