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 7 times and lost 3 times.

1. Goat
2. Goat
3. Car
4. Car
5. Car
6. Goat
7. Car
8. Car
9. Car
10. Car

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 331 times and lost 669 times.

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