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. Goat
7. Goat
8. Car
9. Car
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 337 times and lost 663 times.

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