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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
8. Car
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 336 times and lost 664 times.

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