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 4 times and lost 6 times.

1. Car
2. Car
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
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 341 times and lost 659 times.

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