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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
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 347 times and lost 653 times.

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