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 2 times and lost 8 times.

1. Goat
2. Goat
3. Car
4. Goat
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 342 times and lost 658 times.

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