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 5 times and lost 5 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Car
7. Car
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 354 times and lost 646 times.

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