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. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
10. Car

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 338 times and lost 662 times.

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