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. Car
5. Car
6. Goat
7. Car
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 329 times and lost 671 times.

1. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
9. Goat
10. Car
11. Car
12. Car
13. Goat
14. Goat
15. Goat
16. Goat
17. Goat
18. Goat
19. Car
20. Goat
21. Car
22. Goat
23. Goat
24. Car
25. Goat
26. Goat
27. Car
28. Goat
29. Goat
30. Goat
31. Goat
32. Goat
33. Car
34. Car
35. Goat
36. Car
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Car
43. Car
44. Goat
45. Car
46. Goat
47. Car
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Car
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Goat
62. Car
63. Goat
64. Goat
65. Goat
66. Car
67. Goat
68. Car
69. Goat
70. Goat
71. Car
72. Car
73. Goat
74. Car
75. Goat
76. Goat
77. Goat
78. Goat
79. Goat
80. Goat
81. Goat
82. Car
83. Car
84. Car
85. Goat
86. Goat
87. Goat
88. Car
89. Goat
90. Car
91. Car
92. Goat
93. Goat
94. Goat
95. Goat
96. Goat
97. Car
98. Goat
99. Goat
100. Car
101. Goat
102. Goat
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Car
110. Goat
111. Goat
112. Car
113. Goat
114. Goat
115. Goat
116. Goat
117. Car
118. Goat
119. Goat
120. Goat
121. Goat
122. Car
123. Car
124. Goat
125. Goat
126. Car
127. Goat
128. Goat
129. Goat
130. Goat
131. Car
132. Goat
133. Goat
134. Car
135. Goat
136. Goat
137. Goat
138. Goat
139. Car
140. Goat
141. Car
142. Car
143. Goat
144. Goat
145. Car
146. Car
147. Car
148. Goat
149. Goat
150. Car
151. Goat
152. Car
153. Goat
154. Car
155. Goat
156. Goat
157. Goat
158. Goat
159. Goat
160. Goat
161. Goat
162. Car
163. Car
164. Goat
165. Car
166. Car
167. Car
168. Goat
169. Goat
170. Goat
171. Car
172. Car
173. Car
174. Goat
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Car
181. Goat
182. Goat
183. Goat
184. Goat
185. Goat
186. Car
187. Goat
188. Car
189. Goat
190. Goat
191. Goat
192. Goat
193. Goat
194. Car
195. Car
196. Car
197. Goat
198. Car
199. Goat
200. Car
201. Goat
202. Car
203. Goat
204. Goat
205. Goat
206. Car
207. Goat
208. Goat
209. Goat
210. Car
211. Car
212. Goat
213. Goat
214. Goat
215. Goat
216. Car
217. Goat
218. Goat
219. Car
220. Car
221. Car
222. Car
223. Goat
224. Goat
225. Goat
226. Car
227. Goat
228. Goat
229. Goat
230. Car
231. Goat
232. Car
233. Car
234. Goat
235. Car
236. Goat
237. Goat
238. Car
239. Goat
240. Car
241. Goat
242. Goat
243. Goat
244. Car
245. Goat
246. Car
247. Car
248. Car
249. Car
250. Goat
251. Car
252. Goat
253. Goat
254. Car
255. Goat
256. Car
257. Goat
258. Goat
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Goat
265. Goat
266. Goat
267. Goat
268. Goat
269. Car
270. Goat
271. Car
272. Goat
273. Car
274. Goat
275. Goat
276. Car
277. Goat
278. Car
279. Car
280. Goat
281. Goat
282. Car
283. Goat
284. Car
285. Goat
286. Goat
287. Goat
288. Goat
289. Goat
290. Car
291. Goat
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. Goat
305. Goat
306. Goat
307. Goat
308. Car
309. Goat
310. Goat
311. Goat
312. Goat
313. Goat
314. Goat
315. Car
316. Car
317. Goat
318. Goat
319. Car
320. Car
321. Goat
322. Goat
323. Goat
324. Goat
325. Car
326. Goat
327. Goat
328. Goat
329. Car
330. Car
331. Goat
332. Car
333. Goat
334. Goat
335. Goat
336. Goat
337. Goat
338. Goat
339. Goat
340. Car
341. Goat
342. Goat
343. Car
344. Goat
345. Car
346. Car
347. Goat
348. Car
349. Car
350. Car
351. Car
352. Goat
353. Goat
354. Goat
355. Car
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Goat
365. Car
366. Goat
367. Goat
368. Car
369. Goat
370. Goat
371. Car
372. Goat
373. Goat
374. Goat
375. Goat
376. Car
377. Goat
378. Goat
379. Car
380. Goat
381. Goat
382. Car
383. Car
384. Goat
385. Car
386. Car
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Car
393. Goat
394. Car
395. Goat
396. Goat
397. Car
398. Goat
399. Goat
400. Goat
401. Goat
402. Goat
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Goat
409. Car
410. Goat
411. Goat
412. Car
413. Goat
414. Goat
415. Car
416. Goat
417. Goat
418. Car
419. Car
420. Goat
421. Car
422. Goat
423. Goat
424. Car
425. Goat
426. Goat
427. Car
428. Goat
429. Goat
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Goat
437. Goat
438. Goat
439. Car
440. Car
441. Goat
442. Goat
443. Goat
444. Goat
445. Car
446. Goat
447. Goat
448. Car
449. Goat
450. Goat
451. Goat
452. Goat
453. Car
454. Car
455. Goat
456. Goat
457. Car
458. Goat
459. Car
460. Goat
461. Car
462. Car
463. Car
464. Goat
465. Goat
466. Car
467. Goat
468. Goat
469. Goat
470. Goat
471. Car
472. Goat
473. Goat
474. Car
475. Car
476. Goat
477. Car
478. Goat
479. Goat
480. Goat
481. Car
482. Goat
483. Goat
484. Goat
485. Goat
486. Goat
487. Car
488. Goat
489. Goat
490. Goat
491. Goat
492. Goat
493. Car
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Car
501. Goat
502. Goat
503. Goat
504. Goat
505. Goat
506. Goat
507. Car
508. Car
509. Goat
510. Goat
511. Car
512. Goat
513. Goat
514. Goat
515. Car
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. Car
529. Car
530. Goat
531. Goat
532. Goat
533. Goat
534. Goat
535. Goat
536. Car
537. Goat
538. Car
539. Car
540. Goat
541. Car
542. Car
543. Goat
544. Goat
545. Car
546. Car
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Goat
553. Car
554. Car
555. Car
556. Goat
557. Car
558. Car
559. Goat
560. Goat
561. Car
562. Car
563. Car
564. Goat
565. Goat
566. Car
567. Goat
568. Goat
569. Car
570. Goat
571. Goat
572. Goat
573. Goat
574. Goat
575. Goat
576. Car
577. Goat
578. Car
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Car
592. Goat
593. Goat
594. Goat
595. Car
596. Goat
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. Car
612. Goat
613. Goat
614. Car
615. Goat
616. Goat
617. Car
618. Goat
619. Goat
620. Car
621. Goat
622. Car
623. Goat
624. Goat
625. Car
626. Goat
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Car
633. Goat
634. Car
635. Goat
636. Car
637. Car
638. Car
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Car
645. Car
646. Goat
647. Car
648. Car
649. Car
650. Car
651. Car
652. Goat
653. Goat
654. Goat
655. Goat
656. Goat
657. Car
658. Goat
659. Goat
660. Goat
661. Car
662. Goat
663. Goat
664. Goat
665. Car
666. Goat
667. Car
668. Car
669. Car
670. Goat
671. Car
672. Goat
673. Goat
674. Car
675. Car
676. Goat
677. Goat
678. Car
679. Car
680. Goat
681. Car
682. Goat
683. Car
684. Goat
685. Goat
686. Goat
687. Car
688. Goat
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Car
698. Goat
699. Car
700. Goat
701. Car
702. Car
703. Goat
704. Goat
705. Goat
706. Car
707. Goat
708. Goat
709. Goat
710. Car
711. Goat
712. Goat
713. Car
714. Car
715. Goat
716. Goat
717. Goat
718. Car
719. Car
720. Car
721. Goat
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Goat
728. Goat
729. Car
730. Goat
731. Car
732. Goat
733. Goat
734. Car
735. Goat
736. Goat
737. Goat
738. Goat
739. Car
740. Car
741. Goat
742. Goat
743. Car
744. Car
745. Car
746. Goat
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Car
753. Goat
754. Goat
755. Car
756. Car
757. Goat
758. Goat
759. Goat
760. Goat
761. Goat
762. Car
763. Goat
764. Goat
765. Goat
766. Goat
767. Car
768. Goat
769. Goat
770. Goat
771. Car
772. Goat
773. Goat
774. Goat
775. Goat
776. Car
777. Goat
778. Goat
779. Car
780. Goat
781. Goat
782. Goat
783. Goat
784. Goat
785. Car
786. Goat
787. Car
788. Goat
789. Car
790. Car
791. Car
792. Car
793. Car
794. Goat
795. Goat
796. Car
797. Car
798. Goat
799. Goat
800. Car
801. Goat
802. Car
803. Goat
804. Goat
805. Goat
806. Goat
807. Goat
808. Car
809. Car
810. Goat
811. Goat
812. Car
813. Car
814. Goat
815. Goat
816. Goat
817. Car
818. Car
819. Goat
820. Car
821. Car
822. Goat
823. Goat
824. Goat
825. Goat
826. Goat
827. Car
828. Car
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Car
835. Goat
836. Car
837. Car
838. Goat
839. Goat
840. Car
841. Goat
842. Goat
843. Goat
844. Goat
845. Goat
846. Goat
847. Car
848. Car
849. Goat
850. Car
851. Car
852. Goat
853. Car
854. Goat
855. Goat
856. Car
857. Goat
858. Car
859. Goat
860. Car
861. Goat
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Car
872. Goat
873. Goat
874. Goat
875. Car
876. Goat
877. Goat
878. Car
879. Goat
880. Car
881. Car
882. Car
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Goat
889. Goat
890. Car
891. Car
892. Goat
893. Goat
894. Goat
895. Car
896. Goat
897. Car
898. Goat
899. Goat
900. Goat
901. Goat
902. Car
903. Goat
904. Goat
905. Goat
906. Goat
907. Goat
908. Car
909. Goat
910. Goat
911. Goat
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Goat
924. Goat
925. Goat
926. Goat
927. Goat
928. Car
929. Car
930. Goat
931. Goat
932. Car
933. Goat
934. Goat
935. Goat
936. Car
937. Goat
938. Goat
939. Goat
940. Goat
941. Car
942. Goat
943. Car
944. Goat
945. Goat
946. Car
947. Goat
948. Goat
949. Goat
950. Goat
951. Goat
952. Goat
953. Goat
954. Goat
955. Goat
956. Car
957. Goat
958. Car
959. Goat
960. Goat
961. Goat
962. Goat
963. Goat
964. Goat
965. Goat
966. Goat
967. Goat
968. Car
969. Goat
970. Goat
971. Car
972. Goat
973. Goat
974. Goat
975. Car
976. Goat
977. Goat
978. Goat
979. Car
980. Goat
981. Car
982. Goat
983. Goat
984. Car
985. Goat
986. Car
987. Goat
988. Goat
989. Goat
990. Car
991. Goat
992. Goat
993. Car
994. Goat
995. Car
996. Car
997. Car
998. Goat
999. Car
1000. Car

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 111 times and lost 889 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 30 times and lost 70 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;
	}
}