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 4 times and lost 6 times.

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

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