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. Car
7. Goat
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 355 times and lost 645 times.

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