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. Car
2. Car
3. Goat
4. Car
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 355 times and lost 645 times.

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