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. Goat
5. Car
6. Car
7. Car
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 306 times and lost 694 times.

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