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. Goat
7. Car
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 349 times and lost 651 times.

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