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. Car
5. Goat
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 321 times and lost 679 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Car
11. Goat
12. Car
13. Car
14. Goat
15. Goat
16. Goat
17. Goat
18. Car
19. Goat
20. Goat
21. Goat
22. Goat
23. Car
24. Car
25. Goat
26. Goat
27. Goat
28. Goat
29. Goat
30. Car
31. Goat
32. Goat
33. Car
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Car
44. Goat
45. Goat
46. Car
47. Goat
48. Goat
49. Goat
50. Goat
51. Car
52. Goat
53. Goat
54. Goat
55. Car
56. Goat
57. Car
58. Goat
59. Goat
60. Car
61. Goat
62. Goat
63. Car
64. Goat
65. Goat
66. Goat
67. Goat
68. Goat
69. Car
70. Goat
71. Goat
72. Car
73. Car
74. Goat
75. Car
76. Car
77. Goat
78. Goat
79. Goat
80. Car
81. Car
82. Goat
83. Goat
84. Car
85. Goat
86. Car
87. Goat
88. Car
89. Goat
90. Goat
91. Goat
92. Goat
93. Goat
94. Goat
95. Goat
96. Car
97. Goat
98. Goat
99. Car
100. Car
101. Goat
102. Goat
103. Goat
104. Car
105. Goat
106. Goat
107. Goat
108. Goat
109. Car
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Goat
118. Car
119. Car
120. Goat
121. Car
122. Car
123. Car
124. Goat
125. Goat
126. Car
127. Goat
128. Car
129. Goat
130. Car
131. Goat
132. Car
133. Car
134. Car
135. Car
136. Goat
137. Car
138. Goat
139. Goat
140. Goat
141. Goat
142. Car
143. Goat
144. Goat
145. Car
146. Goat
147. Car
148. Goat
149. Car
150. Goat
151. Car
152. Goat
153. Goat
154. Car
155. Goat
156. Car
157. Goat
158. Goat
159. Goat
160. Goat
161. Goat
162. Car
163. Car
164. Goat
165. Car
166. Goat
167. Goat
168. Goat
169. Goat
170. Goat
171. Goat
172. Goat
173. Goat
174. Car
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Goat
181. Goat
182. Car
183. Goat
184. Goat
185. Goat
186. Goat
187. Goat
188. Goat
189. Goat
190. Goat
191. Car
192. Goat
193. Car
194. Car
195. Goat
196. Goat
197. Car
198. Goat
199. Goat
200. Goat
201. Goat
202. Car
203. Goat
204. Car
205. Goat
206. Goat
207. Car
208. Goat
209. Car
210. Car
211. Car
212. Goat
213. Car
214. Car
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Goat
222. Car
223. Goat
224. Goat
225. Car
226. Car
227. Goat
228. Goat
229. Car
230. Car
231. Car
232. Car
233. Goat
234. Goat
235. Goat
236. Goat
237. Goat
238. Goat
239. Goat
240. Car
241. Goat
242. Goat
243. Goat
244. Goat
245. Goat
246. Goat
247. Car
248. Goat
249. Goat
250. Car
251. Car
252. Goat
253. Goat
254. Goat
255. Car
256. Goat
257. Goat
258. Goat
259. Goat
260. Car
261. Goat
262. Goat
263. Goat
264. Goat
265. Car
266. Goat
267. Car
268. Car
269. Goat
270. Goat
271. Goat
272. Goat
273. Goat
274. Goat
275. Car
276. Goat
277. Car
278. Car
279. Goat
280. Car
281. Goat
282. Goat
283. Goat
284. Goat
285. Goat
286. Car
287. Goat
288. Car
289. Car
290. Car
291. Goat
292. Goat
293. Car
294. Goat
295. Goat
296. Goat
297. Car
298. Car
299. Goat
300. Goat
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Goat
307. Goat
308. Car
309. Goat
310. Goat
311. Goat
312. Car
313. Goat
314. Goat
315. Goat
316. Goat
317. Goat
318. Goat
319. Goat
320. Goat
321. Goat
322. Car
323. Goat
324. Goat
325. Car
326. Goat
327. Goat
328. Car
329. Goat
330. Goat
331. Goat
332. Goat
333. Goat
334. Goat
335. Car
336. Goat
337. Car
338. Goat
339. Goat
340. Goat
341. Goat
342. Goat
343. Car
344. Car
345. Goat
346. Goat
347. Car
348. Goat
349. Goat
350. Goat
351. Goat
352. Car
353. Goat
354. Goat
355. Goat
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Goat
362. Car
363. Car
364. Goat
365. Goat
366. Car
367. Goat
368. Goat
369. Car
370. Goat
371. Goat
372. Goat
373. Car
374. Goat
375. Goat
376. Goat
377. Goat
378. Goat
379. Goat
380. Car
381. Goat
382. Car
383. Goat
384. Goat
385. Car
386. Goat
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Car
403. Goat
404. Goat
405. Goat
406. Goat
407. Goat
408. Goat
409. Car
410. Goat
411. Goat
412. Car
413. Goat
414. Goat
415. Goat
416. Car
417. Goat
418. Goat
419. Goat
420. Goat
421. Goat
422. Car
423. Car
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Car
431. Car
432. Goat
433. Goat
434. Goat
435. Car
436. Car
437. Goat
438. Goat
439. Car
440. Goat
441. Car
442. Goat
443. Car
444. Car
445. Goat
446. Car
447. Goat
448. Car
449. Goat
450. Goat
451. Goat
452. Goat
453. Goat
454. Car
455. Car
456. Goat
457. Goat
458. Goat
459. Car
460. Car
461. Goat
462. Goat
463. Goat
464. Car
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Goat
471. Car
472. Car
473. Car
474. Goat
475. Goat
476. Car
477. Goat
478. Goat
479. Goat
480. Goat
481. Goat
482. Goat
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Car
492. Car
493. Car
494. Car
495. Goat
496. Goat
497. Goat
498. Car
499. Car
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. Car
513. Goat
514. Car
515. Goat
516. Goat
517. Goat
518. Goat
519. Goat
520. Goat
521. Car
522. Goat
523. Car
524. Goat
525. Goat
526. Goat
527. Car
528. Goat
529. Goat
530. Goat
531. Goat
532. Goat
533. Car
534. Car
535. Car
536. Goat
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Goat
545. Car
546. Car
547. Car
548. Goat
549. Car
550. Goat
551. Goat
552. Goat
553. Goat
554. Car
555. Car
556. Goat
557. Goat
558. Goat
559. Goat
560. Goat
561. Goat
562. Car
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Goat
570. Goat
571. Goat
572. Goat
573. Goat
574. Car
575. Goat
576. Goat
577. Goat
578. Car
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Car
587. Goat
588. Goat
589. Goat
590. Car
591. Goat
592. Goat
593. Car
594. Goat
595. Car
596. Goat
597. Goat
598. Goat
599. Goat
600. Car
601. Goat
602. Goat
603. Goat
604. Car
605. Car
606. Goat
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Goat
614. Goat
615. Goat
616. Goat
617. Goat
618. Car
619. Goat
620. Car
621. Car
622. Car
623. Car
624. Goat
625. Goat
626. Car
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Car
635. Goat
636. Goat
637. Goat
638. Car
639. Goat
640. Car
641. Goat
642. Car
643. Car
644. Car
645. Goat
646. Goat
647. Car
648. Car
649. Goat
650. Goat
651. Goat
652. Car
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Car
659. Goat
660. Goat
661. Car
662. Goat
663. Goat
664. Goat
665. Car
666. Car
667. Goat
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Car
674. Goat
675. Car
676. Car
677. Goat
678. Car
679. Goat
680. Goat
681. Goat
682. Goat
683. Car
684. Car
685. Goat
686. Car
687. Goat
688. Goat
689. Car
690. Car
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Goat
698. Goat
699. Goat
700. Goat
701. Goat
702. Goat
703. Goat
704. Car
705. Goat
706. Car
707. Goat
708. Car
709. Car
710. Goat
711. Car
712. Goat
713. Goat
714. Goat
715. Car
716. Goat
717. Goat
718. Goat
719. Goat
720. Car
721. Goat
722. Goat
723. Car
724. Car
725. Car
726. Car
727. Goat
728. Goat
729. Car
730. Goat
731. Car
732. Car
733. Car
734. Goat
735. Car
736. Goat
737. Car
738. Car
739. Car
740. Car
741. Goat
742. Goat
743. Car
744. Goat
745. Goat
746. Car
747. Goat
748. Car
749. Car
750. Goat
751. Goat
752. Goat
753. Goat
754. Goat
755. Car
756. Car
757. Goat
758. Goat
759. Goat
760. Goat
761. Car
762. Goat
763. Goat
764. Goat
765. Goat
766. Goat
767. Goat
768. Goat
769. Car
770. Goat
771. Goat
772. Goat
773. Car
774. Goat
775. Goat
776. Goat
777. Goat
778. Car
779. Goat
780. Goat
781. Goat
782. Goat
783. Goat
784. Goat
785. Goat
786. Goat
787. Car
788. Goat
789. Goat
790. Car
791. Car
792. Goat
793. Goat
794. Car
795. Goat
796. Car
797. Car
798. Goat
799. Car
800. Goat
801. Goat
802. Car
803. Car
804. Car
805. Goat
806. Goat
807. Goat
808. Goat
809. Car
810. Car
811. Car
812. Car
813. Car
814. Goat
815. Car
816. Car
817. Car
818. Goat
819. Car
820. Goat
821. Car
822. Car
823. Car
824. Goat
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Car
833. Goat
834. Goat
835. Goat
836. Car
837. Goat
838. Goat
839. Car
840. Goat
841. Car
842. Goat
843. Goat
844. Car
845. Car
846. Goat
847. Car
848. Goat
849. Goat
850. Car
851. Goat
852. Goat
853. Car
854. Car
855. Goat
856. Goat
857. Goat
858. Goat
859. Car
860. Car
861. Goat
862. Goat
863. Goat
864. Car
865. Car
866. Goat
867. Goat
868. Goat
869. Car
870. Goat
871. Goat
872. Goat
873. Car
874. Goat
875. Car
876. Car
877. Car
878. Goat
879. Goat
880. Goat
881. Car
882. Goat
883. Goat
884. Goat
885. Goat
886. Goat
887. Car
888. Goat
889. Goat
890. Car
891. Goat
892. Goat
893. Car
894. Goat
895. Goat
896. Car
897. Car
898. Goat
899. Car
900. Car
901. Car
902. Goat
903. Goat
904. Car
905. Goat
906. Goat
907. Goat
908. Car
909. Goat
910. Goat
911. Car
912. Car
913. Car
914. Goat
915. Car
916. Goat
917. Goat
918. Goat
919. Goat
920. Car
921. Goat
922. Goat
923. Car
924. Car
925. Car
926. Car
927. Car
928. Goat
929. Goat
930. Goat
931. Goat
932. Goat
933. Goat
934. Car
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Goat
941. Goat
942. Car
943. Goat
944. Goat
945. Goat
946. Goat
947. Goat
948. Goat
949. Car
950. Goat
951. Car
952. Goat
953. Goat
954. Goat
955. Goat
956. Car
957. Goat
958. Car
959. Goat
960. Car
961. Car
962. Goat
963. Goat
964. Car
965. Car
966. Car
967. Goat
968. Goat
969. Goat
970. Car
971. Goat
972. Goat
973. Car
974. Goat
975. Car
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Car
983. Goat
984. Car
985. Goat
986. Goat
987. Car
988. Goat
989. Goat
990. Car
991. Goat
992. Goat
993. Goat
994. Goat
995. Goat
996. Car
997. Goat
998. Goat
999. Goat
1000. Car

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 113 times and lost 887 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;
	}
}