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. Car
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Car
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 311 times and lost 689 times.

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