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 0 time and lost 10 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
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 324 times and lost 676 times.

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