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 7 times and lost 3 times.

1. Goat
2. Car
3. Car
4. Car
5. Car
6. Car
7. Goat
8. Goat
9. Car
10. Car

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 341 times and lost 659 times.

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