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 5 times and lost 5 times.

1. Car
2. Goat
3. Car
4. Car
5. Car
6. Car
7. Goat
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 362 times and lost 638 times.

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