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. Goat
4. Car
5. Car
6. Car
7. Goat
8. Goat
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 337 times and lost 663 times.

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