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. Car
2. Car
3. Car
4. Goat
5. Goat
6. Car
7. Car
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 322 times and lost 678 times.

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