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. Goat
2. Car
3. Car
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 325 times and lost 675 times.

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