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 3 times and lost 7 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Car
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 342 times and lost 658 times.

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