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. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Car
8. Goat
9. Car
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 327 times and lost 673 times.

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