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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
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 340 times and lost 660 times.

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