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 3 times and lost 7 times.

1. Goat
2. Goat
3. Car
4. Car
5. Car
6. Goat
7. Goat
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 327 times and lost 673 times.

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