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. Car
2. Goat
3. Goat
4. Goat
5. Goat
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 346 times and lost 654 times.

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