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 4 times and lost 6 times.

1. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
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 335 times and lost 665 times.

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