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 6 times and lost 4 times.

1. Goat
2. Goat
3. Car
4. Car
5. Car
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 315 times and lost 685 times.

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