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. Car
4. Goat
5. Car
6. Car
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 348 times and lost 652 times.

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