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 8 times and lost 2 times.

1. Goat
2. Goat
3. Car
4. Car
5. Car
6. Car
7. Car
8. Car
9. Car
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 317 times and lost 683 times.

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