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 7 times and lost 3 times.

1. Goat
2. Car
3. Car
4. Goat
5. Car
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 328 times and lost 672 times.

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