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 5 times and lost 5 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Car
8. Car
9. Car
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 344 times and lost 656 times.

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