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 0 time and lost 10 times.

1. Goat
2. Goat
3. Goat
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 340 times and lost 660 times.

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