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. Car
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
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 353 times and lost 647 times.

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