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. Goat
2. Car
3. Car
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
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 331 times and lost 669 times.

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