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 5 times and lost 5 times.

1. Car
2. Car
3. Car
4. Car
5. Goat
6. Goat
7. Car
8. Goat
9. Goat
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 336 times and lost 664 times.

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