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. Car
5. Goat
6. Goat
7. Goat
8. Car
9. Goat
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 327 times and lost 673 times.

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