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 7 times and lost 3 times.

1. Goat
2. Car
3. Goat
4. Car
5. Car
6. Car
7. Car
8. Car
9. Goat
10. Car

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 315 times and lost 685 times.

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