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. Car
7. Goat
8. Goat
9. Goat
10. Car

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 314 times and lost 686 times.

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