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. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Car
9. Car
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 360 times and lost 640 times.

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