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. Goat
6. Goat
7. Car
8. Goat
9. Car
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 341 times and lost 659 times.

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