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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Car
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 310 times and lost 690 times.

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