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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Car
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 320 times and lost 680 times.

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