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. Goat
2. Goat
3. Car
4. Car
5. Goat
6. Car
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 333 times and lost 667 times.

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