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 1 time and lost 9 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat

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 350 times and lost 650 times.

1. Goat
2. Goat
3. Car
4. Car
5. Car
6. Car
7. Car
8. Car
9. Car
10. Goat
11. Goat
12. Goat
13. Goat
14. Goat
15. Car
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Car
24. Goat
25. Goat
26. Car
27. Goat
28. Goat
29. Goat
30. Car
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Car
37. Car
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Car
46. Car
47. Car
48. Goat
49. Goat
50. Car
51. Car
52. Car
53. Car
54. Car
55. Car
56. Car
57. Car
58. Car
59. Car
60. Goat
61. Car
62. Car
63. Car
64. Goat
65. Goat
66. Goat
67. Car
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Goat
78. Goat
79. Car
80. Goat
81. Goat
82. Goat
83. Goat
84. Car
85. Car
86. Goat
87. Goat
88. Goat
89. Car
90. Goat
91. Goat
92. Goat
93. Car
94. Goat
95. Goat
96. Goat
97. Goat
98. Goat
99. Goat
100. Car
101. Goat
102. Goat
103. Car
104. Goat
105. Goat
106. Car
107. Goat
108. Car
109. Car
110. Goat
111. Car
112. Goat
113. Goat
114. Goat
115. Car
116. Goat
117. Goat
118. Goat
119. Goat
120. Car
121. Goat
122. Goat
123. Car
124. Car
125. Car
126. Goat
127. Car
128. Goat
129. Goat
130. Goat
131. Goat
132. Car
133. Goat
134. Goat
135. Car
136. Goat
137. Goat
138. Goat
139. Goat
140. Car
141. Car
142. Goat
143. Goat
144. Goat
145. Car
146. Car
147. Goat
148. Car
149. Car
150. Car
151. Goat
152. Goat
153. Goat
154. Goat
155. Goat
156. Car
157. Car
158. Goat
159. Goat
160. Goat
161. Goat
162. Goat
163. Goat
164. Car
165. Goat
166. Goat
167. Car
168. Car
169. Car
170. Goat
171. Goat
172. Goat
173. Goat
174. Car
175. Car
176. Car
177. Car
178. Car
179. Goat
180. Goat
181. Car
182. Goat
183. Goat
184. Car
185. Goat
186. Car
187. Goat
188. Car
189. Goat
190. Goat
191. Goat
192. Car
193. Car
194. Car
195. Goat
196. Goat
197. Car
198. Goat
199. Goat
200. Goat
201. Car
202. Car
203. Car
204. Goat
205. Goat
206. Car
207. Goat
208. Car
209. Goat
210. Goat
211. Goat
212. Goat
213. Goat
214. Car
215. Car
216. Goat
217. Goat
218. Goat
219. Goat
220. Goat
221. Goat
222. Goat
223. Car
224. Car
225. Goat
226. Car
227. Goat
228. Goat
229. Car
230. Goat
231. Car
232. Car
233. Goat
234. Goat
235. Goat
236. Goat
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Goat
244. Car
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Car
251. Goat
252. Car
253. Car
254. Car
255. Goat
256. Goat
257. Car
258. Goat
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Goat
265. Goat
266. Goat
267. Goat
268. Goat
269. Goat
270. Car
271. Goat
272. Goat
273. Goat
274. Car
275. Goat
276. Goat
277. Goat
278. Goat
279. Goat
280. Goat
281. Goat
282. Car
283. Goat
284. Car
285. Goat
286. Goat
287. Goat
288. Goat
289. Car
290. Car
291. Goat
292. Car
293. Goat
294. Goat
295. Goat
296. Car
297. Goat
298. Car
299. Goat
300. Goat
301. Goat
302. Car
303. Car
304. Goat
305. Goat
306. Goat
307. Goat
308. Goat
309. Car
310. Goat
311. Goat
312. Goat
313. Car
314. Goat
315. Car
316. Goat
317. Car
318. Goat
319. Goat
320. Goat
321. Car
322. Goat
323. Car
324. Car
325. Car
326. Car
327. Car
328. Car
329. Goat
330. Goat
331. Car
332. Car
333. Goat
334. Car
335. Goat
336. Goat
337. Goat
338. Goat
339. Goat
340. Goat
341. Car
342. Goat
343. Goat
344. Goat
345. Goat
346. Car
347. Car
348. Car
349. Goat
350. Car
351. Car
352. Car
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Car
359. Goat
360. Goat
361. Car
362. Goat
363. Goat
364. Car
365. Goat
366. Goat
367. Goat
368. Goat
369. Car
370. Car
371. Car
372. Goat
373. Goat
374. Goat
375. Car
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Goat
382. Goat
383. Goat
384. Car
385. Goat
386. Car
387. Car
388. Goat
389. Goat
390. Goat
391. Car
392. Goat
393. Goat
394. Car
395. Goat
396. Goat
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Goat
403. Goat
404. Goat
405. Goat
406. Goat
407. Goat
408. Car
409. Car
410. Car
411. Car
412. Car
413. Goat
414. Car
415. Goat
416. Goat
417. Goat
418. Car
419. Car
420. Car
421. Goat
422. Goat
423. Car
424. Car
425. Goat
426. Car
427. Car
428. Car
429. Goat
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Car
437. Goat
438. Car
439. Goat
440. Goat
441. Goat
442. Goat
443. Car
444. Goat
445. Goat
446. Goat
447. Goat
448. Goat
449. Car
450. Goat
451. Car
452. Goat
453. Car
454. Car
455. Goat
456. Car
457. Goat
458. Car
459. Car
460. Goat
461. Goat
462. Car
463. Car
464. Car
465. Car
466. Car
467. Car
468. Goat
469. Car
470. Goat
471. Car
472. Goat
473. Car
474. Car
475. Car
476. Goat
477. Goat
478. Goat
479. Goat
480. Car
481. Car
482. Goat
483. Car
484. Car
485. Goat
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Goat
492. Goat
493. Goat
494. Goat
495. Car
496. Goat
497. Goat
498. Goat
499. Car
500. Goat
501. Goat
502. Car
503. Car
504. Car
505. Goat
506. Car
507. Goat
508. Car
509. Goat
510. Goat
511. Goat
512. Goat
513. Car
514. Goat
515. Car
516. Car
517. Goat
518. Car
519. Goat
520. Goat
521. Car
522. Car
523. Goat
524. Car
525. Goat
526. Goat
527. Goat
528. Car
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Car
535. Car
536. Goat
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Car
543. Goat
544. Car
545. Goat
546. Goat
547. Goat
548. Car
549. Goat
550. Car
551. Goat
552. Car
553. Goat
554. Goat
555. Car
556. Goat
557. Car
558. Goat
559. Goat
560. Goat
561. Goat
562. Goat
563. Car
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Car
570. Goat
571. Car
572. Goat
573. Goat
574. Car
575. Goat
576. Goat
577. Car
578. Goat
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Car
585. Goat
586. Goat
587. Car
588. Goat
589. Car
590. Car
591. Goat
592. Car
593. Car
594. Car
595. Goat
596. Goat
597. Goat
598. Goat
599. Car
600. Goat
601. Goat
602. Goat
603. Car
604. Goat
605. Goat
606. Goat
607. Car
608. Goat
609. Car
610. Goat
611. Goat
612. Goat
613. Goat
614. Goat
615. Goat
616. Car
617. Goat
618. Goat
619. Car
620. Goat
621. Goat
622. Car
623. Goat
624. Car
625. Goat
626. Goat
627. Goat
628. Goat
629. Goat
630. Goat
631. Car
632. Goat
633. Goat
634. Car
635. Goat
636. Car
637. Goat
638. Goat
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Goat
645. Car
646. Goat
647. Car
648. Car
649. Car
650. Car
651. Goat
652. Goat
653. Car
654. Car
655. Car
656. Goat
657. Car
658. Car
659. Goat
660. Goat
661. Goat
662. Goat
663. Goat
664. Car
665. Goat
666. Goat
667. Car
668. Goat
669. Goat
670. Goat
671. Car
672. Car
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Car
679. Goat
680. Car
681. Goat
682. Goat
683. Goat
684. Car
685. Goat
686. Car
687. Goat
688. Car
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Car
696. Goat
697. Car
698. Car
699. Goat
700. Car
701. Goat
702. Car
703. Car
704. Car
705. Car
706. Goat
707. Goat
708. Car
709. Car
710. Goat
711. Goat
712. Car
713. Car
714. Goat
715. Goat
716. Car
717. Goat
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Goat
727. Car
728. Goat
729. Car
730. Goat
731. Goat
732. Car
733. Goat
734. Goat
735. Car
736. Goat
737. Goat
738. Goat
739. Goat
740. Goat
741. Goat
742. Goat
743. Goat
744. Goat
745. Goat
746. Goat
747. Goat
748. Goat
749. Goat
750. Goat
751. Goat
752. Goat
753. Car
754. Car
755. Car
756. Car
757. Goat
758. Car
759. Car
760. Car
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Goat
767. Car
768. Car
769. Goat
770. Car
771. Car
772. Goat
773. Goat
774. Car
775. Goat
776. Goat
777. Goat
778. Goat
779. Goat
780. Goat
781. Goat
782. Goat
783. Goat
784. Car
785. Goat
786. Car
787. Car
788. Car
789. Goat
790. Goat
791. Car
792. Car
793. Goat
794. Goat
795. Car
796. Goat
797. Goat
798. Goat
799. Goat
800. Goat
801. Goat
802. Goat
803. Car
804. Car
805. Goat
806. Car
807. Goat
808. Goat
809. Goat
810. Goat
811. Goat
812. Goat
813. Goat
814. Car
815. Car
816. Car
817. Goat
818. Goat
819. Goat
820. Car
821. Car
822. Goat
823. Goat
824. Goat
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Car
833. Goat
834. Goat
835. Car
836. Car
837. Goat
838. Goat
839. Car
840. Car
841. Goat
842. Goat
843. Goat
844. Car
845. Goat
846. Goat
847. Goat
848. Car
849. Car
850. Goat
851. Goat
852. Goat
853. Car
854. Goat
855. Goat
856. Car
857. Goat
858. Goat
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Car
866. Goat
867. Goat
868. Car
869. Goat
870. Car
871. Goat
872. Goat
873. Goat
874. Goat
875. Goat
876. Goat
877. Goat
878. Goat
879. Goat
880. Car
881. Goat
882. Goat
883. Car
884. Goat
885. Goat
886. Goat
887. Car
888. Car
889. Car
890. Goat
891. Car
892. Goat
893. Car
894. Goat
895. Goat
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Goat
905. Goat
906. Car
907. Goat
908. Car
909. Car
910. Goat
911. Car
912. Car
913. Goat
914. Goat
915. Car
916. Car
917. Goat
918. Car
919. Car
920. Goat
921. Car
922. Goat
923. Goat
924. Car
925. Goat
926. Car
927. Goat
928. Goat
929. Car
930. Goat
931. Goat
932. Goat
933. Goat
934. Goat
935. Goat
936. Car
937. Goat
938. Goat
939. Goat
940. Car
941. Car
942. Goat
943. Goat
944. Goat
945. Car
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Car
952. Car
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. Car
966. Goat
967. Goat
968. Car
969. Goat
970. Car
971. Goat
972. Goat
973. Car
974. Car
975. Goat
976. Goat
977. Goat
978. Goat
979. Car
980. Car
981. Goat
982. Goat
983. Goat
984. Car
985. Goat
986. Car
987. Car
988. Goat
989. Car
990. Goat
991. Goat
992. Goat
993. Goat
994. Car
995. Car
996. Goat
997. Car
998. Goat
999. Goat
1000. Car

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 112 times and lost 888 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 28 times and lost 72 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;
	}
}