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. Car
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 354 times and lost 646 times.

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