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 3 times and lost 7 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Car
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 358 times and lost 642 times.

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