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. Car
2. Goat
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Goat
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 354 times and lost 646 times.

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