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. Car
2. Car
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Car
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 327 times and lost 673 times.

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