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 2 times and lost 8 times.

1. Goat
2. Goat
3. Car
4. Goat
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 328 times and lost 672 times.

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