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 7 times and lost 3 times.

1. Car
2. Goat
3. Car
4. Car
5. Car
6. Car
7. Goat
8. Car
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 352 times and lost 648 times.

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