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. Car
3. Car
4. Car
5. Goat
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 347 times and lost 653 times.

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