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. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
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 323 times and lost 677 times.

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