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. Goat
5. Goat
6. Goat
7. Goat
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 342 times and lost 658 times.

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