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. Car
5. Goat
6. Car
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 327 times and lost 673 times.

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