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. Car
3. Car
4. Car
5. Goat
6. Goat
7. Car
8. Car
9. Goat
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 333 times and lost 667 times.

1. Car
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Car
14. Goat
15. Goat
16. Car
17. Goat
18. Car
19. Car
20. Goat
21. Car
22. Goat
23. Goat
24. Goat
25. Car
26. Goat
27. Goat
28. Car
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Car
41. Car
42. Goat
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Goat
62. Goat
63. Car
64. Goat
65. Goat
66. Goat
67. Goat
68. Car
69. Goat
70. Goat
71. Goat
72. Goat
73. Car
74. Goat
75. Goat
76. Car
77. Goat
78. Goat
79. Goat
80. Goat
81. Car
82. Car
83. Goat
84. Goat
85. Goat
86. Car
87. Car
88. Goat
89. Goat
90. Goat
91. Goat
92. Goat
93. Goat
94. Car
95. Goat
96. Car
97. Goat
98. Goat
99. Goat
100. Goat
101. Goat
102. Car
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Car
115. Goat
116. Goat
117. Car
118. Car
119. Car
120. Goat
121. Goat
122. Car
123. Goat
124. Goat
125. Goat
126. Car
127. Car
128. Goat
129. Car
130. Car
131. Car
132. Car
133. Goat
134. Car
135. Car
136. Car
137. Goat
138. Goat
139. Car
140. Car
141. Car
142. Goat
143. Car
144. Goat
145. Goat
146. Goat
147. Car
148. Goat
149. Car
150. Goat
151. Goat
152. Car
153. Goat
154. Goat
155. Car
156. Car
157. Goat
158. Goat
159. Goat
160. Goat
161. Goat
162. Goat
163. Car
164. Goat
165. Goat
166. Car
167. Goat
168. Car
169. Car
170. Goat
171. Goat
172. Goat
173. Goat
174. Car
175. Goat
176. Car
177. Car
178. Car
179. Goat
180. Goat
181. Goat
182. Goat
183. Car
184. Goat
185. Goat
186. Car
187. Car
188. Car
189. Goat
190. Goat
191. Goat
192. Goat
193. Goat
194. Car
195. Goat
196. Goat
197. Car
198. Car
199. Car
200. Car
201. Car
202. Car
203. Goat
204. Car
205. Car
206. Goat
207. Goat
208. Goat
209. Goat
210. Goat
211. Car
212. Goat
213. Goat
214. Goat
215. Car
216. Goat
217. Car
218. Goat
219. Car
220. Goat
221. Goat
222. Car
223. Goat
224. Car
225. Car
226. Goat
227. Car
228. Car
229. Goat
230. Car
231. Goat
232. Goat
233. Goat
234. Car
235. Goat
236. Goat
237. Car
238. Car
239. Goat
240. Car
241. Goat
242. Goat
243. Goat
244. Car
245. Goat
246. Car
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Goat
253. Car
254. Car
255. Goat
256. Goat
257. Goat
258. Goat
259. Goat
260. Goat
261. Goat
262. Goat
263. Goat
264. Car
265. Goat
266. Goat
267. Goat
268. Goat
269. Car
270. Car
271. Goat
272. Goat
273. Goat
274. Car
275. Goat
276. Goat
277. Car
278. Car
279. Car
280. Goat
281. Goat
282. Goat
283. Car
284. Goat
285. Car
286. Goat
287. Goat
288. Car
289. Car
290. Goat
291. Car
292. Goat
293. Goat
294. Goat
295. Goat
296. Goat
297. Goat
298. Goat
299. Goat
300. Goat
301. Car
302. Goat
303. Goat
304. Car
305. Car
306. Goat
307. Goat
308. Goat
309. Goat
310. Car
311. Goat
312. Car
313. Goat
314. Goat
315. Car
316. Goat
317. Goat
318. Car
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. Car
334. Car
335. Car
336. Goat
337. Car
338. Goat
339. Goat
340. Goat
341. Goat
342. Goat
343. Goat
344. Goat
345. Car
346. Car
347. Goat
348. Goat
349. Car
350. Goat
351. Goat
352. Goat
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Goat
359. Goat
360. Car
361. Goat
362. Car
363. Goat
364. Car
365. Car
366. Goat
367. Goat
368. Car
369. Goat
370. Car
371. Car
372. Goat
373. Goat
374. Car
375. Car
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Goat
382. Car
383. Goat
384. Goat
385. Goat
386. Car
387. Goat
388. Goat
389. Car
390. Car
391. Goat
392. Car
393. Car
394. Goat
395. Car
396. Goat
397. Goat
398. Car
399. Goat
400. Car
401. Car
402. Car
403. Car
404. Car
405. Goat
406. Goat
407. Car
408. Goat
409. Car
410. Goat
411. Goat
412. Car
413. Goat
414. Goat
415. Goat
416. Goat
417. Car
418. Goat
419. Car
420. Goat
421. Goat
422. Goat
423. Goat
424. Car
425. Goat
426. Goat
427. Goat
428. Car
429. Car
430. Car
431. Car
432. Goat
433. Goat
434. Goat
435. Goat
436. Car
437. Goat
438. Goat
439. Car
440. Goat
441. Goat
442. Car
443. Goat
444. Car
445. Goat
446. Car
447. Car
448. Goat
449. Car
450. Goat
451. Goat
452. Goat
453. Goat
454. Car
455. Car
456. Car
457. Car
458. Car
459. Goat
460. Goat
461. Goat
462. Goat
463. Goat
464. Car
465. Goat
466. Goat
467. Car
468. Car
469. Goat
470. Car
471. Car
472. Car
473. Car
474. Car
475. Goat
476. Goat
477. Car
478. Car
479. Car
480. Car
481. Goat
482. Goat
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Goat
492. Goat
493. Car
494. Goat
495. Car
496. Car
497. Goat
498. Goat
499. Goat
500. Goat
501. Car
502. Goat
503. Goat
504. Car
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. Goat
518. Car
519. Goat
520. Car
521. Car
522. Goat
523. Goat
524. Goat
525. Goat
526. Car
527. Goat
528. Car
529. Goat
530. Car
531. Goat
532. Goat
533. Car
534. Goat
535. Goat
536. Goat
537. Goat
538. Goat
539. Goat
540. Goat
541. Car
542. Goat
543. Goat
544. Goat
545. Goat
546. Goat
547. Car
548. Goat
549. Goat
550. Car
551. Goat
552. Car
553. Car
554. Goat
555. Goat
556. Car
557. Car
558. Goat
559. Car
560. Car
561. Car
562. Goat
563. Car
564. Car
565. Goat
566. Goat
567. Goat
568. Car
569. Car
570. Goat
571. Goat
572. Goat
573. Goat
574. Car
575. Car
576. Car
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Car
583. Goat
584. Car
585. Car
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Goat
592. Car
593. Car
594. Goat
595. Goat
596. Car
597. Goat
598. Goat
599. Goat
600. Goat
601. Goat
602. Car
603. Goat
604. Goat
605. Goat
606. Goat
607. Car
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Goat
614. Goat
615. Car
616. Goat
617. Goat
618. Goat
619. Goat
620. Goat
621. Car
622. Goat
623. Goat
624. Car
625. Goat
626. Car
627. Goat
628. Car
629. Goat
630. Goat
631. Goat
632. Goat
633. Car
634. Goat
635. Goat
636. Goat
637. Goat
638. Goat
639. Car
640. Goat
641. Goat
642. Goat
643. Goat
644. Car
645. Goat
646. Goat
647. Car
648. Goat
649. Goat
650. Goat
651. Car
652. Goat
653. Car
654. Goat
655. Car
656. Car
657. Goat
658. Goat
659. Goat
660. Goat
661. Goat
662. Goat
663. Goat
664. Car
665. Car
666. Goat
667. Car
668. Goat
669. Car
670. Goat
671. Car
672. Goat
673. Car
674. Goat
675. Goat
676. Car
677. Goat
678. Goat
679. Car
680. Car
681. Car
682. Car
683. Goat
684. Goat
685. Goat
686. Goat
687. Goat
688. Car
689. Goat
690. Goat
691. Car
692. Goat
693. Car
694. Car
695. Car
696. Goat
697. Goat
698. Goat
699. Goat
700. Goat
701. Goat
702. Goat
703. Goat
704. Goat
705. Goat
706. Car
707. Goat
708. Car
709. Goat
710. Car
711. Car
712. Car
713. Goat
714. Car
715. Goat
716. Goat
717. Car
718. Goat
719. Car
720. Car
721. Goat
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Car
728. Goat
729. Goat
730. Car
731. Goat
732. Car
733. Car
734. Goat
735. Goat
736. Goat
737. Car
738. Goat
739. Goat
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Goat
746. Car
747. Car
748. Goat
749. Car
750. Car
751. Goat
752. Goat
753. Goat
754. Car
755. Goat
756. Car
757. Goat
758. Goat
759. Goat
760. Goat
761. Car
762. Car
763. Goat
764. Car
765. Car
766. Goat
767. Goat
768. Goat
769. Car
770. Goat
771. Car
772. Goat
773. Car
774. Goat
775. Goat
776. Goat
777. Car
778. Goat
779. Goat
780. Car
781. Goat
782. Car
783. Car
784. Goat
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Goat
792. Goat
793. Goat
794. Car
795. Goat
796. Car
797. Goat
798. Goat
799. Goat
800. Goat
801. Goat
802. Goat
803. Car
804. Goat
805. Goat
806. Goat
807. Goat
808. Goat
809. Car
810. Car
811. Goat
812. Car
813. Goat
814. Goat
815. Car
816. Goat
817. Goat
818. Goat
819. Car
820. Goat
821. Goat
822. Goat
823. Car
824. Car
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Goat
836. Goat
837. Car
838. Goat
839. Goat
840. Car
841. Goat
842. Goat
843. Goat
844. Goat
845. Goat
846. Car
847. Goat
848. Car
849. Car
850. Car
851. Goat
852. Car
853. Goat
854. Goat
855. Goat
856. Car
857. Goat
858. Goat
859. Goat
860. Goat
861. Goat
862. Car
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Car
874. Goat
875. Goat
876. Goat
877. Car
878. Car
879. Goat
880. Goat
881. Goat
882. Goat
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Goat
889. Goat
890. Goat
891. Car
892. Goat
893. Goat
894. Goat
895. Goat
896. Car
897. Goat
898. Goat
899. Goat
900. Car
901. Goat
902. Car
903. Goat
904. Goat
905. Goat
906. Car
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Goat
913. Car
914. Goat
915. Goat
916. Goat
917. Goat
918. Car
919. Goat
920. Goat
921. Car
922. Goat
923. Car
924. Goat
925. Goat
926. Goat
927. Car
928. Goat
929. Car
930. Goat
931. Car
932. Goat
933. Car
934. Goat
935. Goat
936. Car
937. Car
938. Goat
939. Goat
940. Goat
941. Car
942. Goat
943. Goat
944. Goat
945. Goat
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Goat
952. Car
953. Goat
954. Goat
955. Goat
956. Goat
957. Goat
958. Car
959. Car
960. Goat
961. Goat
962. Goat
963. Goat
964. Car
965. Goat
966. Goat
967. Car
968. Car
969. Goat
970. Goat
971. Goat
972. Car
973. Goat
974. Car
975. Goat
976. Car
977. Goat
978. Goat
979. Car
980. Goat
981. Goat
982. Car
983. Goat
984. Car
985. Car
986. Goat
987. Car
988. Goat
989. Car
990. Goat
991. Goat
992. Car
993. Goat
994. Goat
995. Goat
996. Goat
997. Goat
998. Car
999. Car
1000. Goat

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 77 times and lost 923 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 37 times and lost 63 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;
	}
}