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. Car
3. Goat
4. Car
5. Car
6. Goat
7. Goat
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 354 times and lost 646 times.

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