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 3 times and lost 7 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Car
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 327 times and lost 673 times.

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