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 1 time and lost 9 times.

1. Car
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 333 times and lost 667 times.

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