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. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Car
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 340 times and lost 660 times.

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