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. Goat
2. Car
3. Car
4. Goat
5. Car
6. Goat
7. Car
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 337 times and lost 663 times.

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