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 2 times and lost 8 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
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 330 times and lost 670 times.

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