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. Car
3. Car
4. Car
5. Car
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 340 times and lost 660 times.

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