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 4 times and lost 6 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Car
7. Car
8. Goat
9. Car
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 350 times and lost 650 times.

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