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 3 times and lost 7 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Car
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 327 times and lost 673 times.

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