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 4 times and lost 6 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
8. Car
9. Car
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 361 times and lost 639 times.

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