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 2 times and lost 8 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
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 337 times and lost 663 times.

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