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 6 times and lost 4 times.

1. Car
2. Goat
3. Goat
4. Car
5. Car
6. Car
7. Car
8. Car
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 344 times and lost 656 times.

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