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 2 times and lost 8 times.

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