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. Car
3. Goat
4. Goat
5. Goat
6. Car
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 335 times and lost 665 times.

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