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. Car
4. Goat
5. Car
6. Car
7. Goat
8. Car
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 339 times and lost 661 times.

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