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. Car
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 337 times and lost 663 times.

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