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 5 times and lost 5 times.

1. Goat
2. Car
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Car
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 347 times and lost 653 times.

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