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. Car
3. Goat
4. Car
5. Car
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 330 times and lost 670 times.

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