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. Car
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
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 326 times and lost 674 times.

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