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. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
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 331 times and lost 669 times.

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