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. Car
4. Goat
5. Goat
6. Car
7. Car
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 318 times and lost 682 times.

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