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. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Goat
9. Car
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 342 times and lost 658 times.

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