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. Car
3. Goat
4. Goat
5. Car
6. Goat
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 383 times and lost 617 times.

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