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. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Car
7. Car
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 357 times and lost 643 times.

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