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