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 6 times and lost 4 times.

1. Car
2. Goat
3. Car
4. Car
5. Car
6. Goat
7. Goat
8. Car
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 332 times and lost 668 times.

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