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. Car
3. Goat
4. Goat
5. Goat
6. Car
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 354 times and lost 646 times.

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