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 3 times and lost 7 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Car
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 326 times and lost 674 times.

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