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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
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 349 times and lost 651 times.

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