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. Goat
6. Car
7. Car
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 304 times and lost 696 times.

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