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. Goat
7. Goat
8. Goat
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 338 times and lost 662 times.

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