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. Goat
7. Goat
8. Goat
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 348 times and lost 652 times.

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