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. Goat
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Goat
10. Goat

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 339 times and lost 661 times.

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