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. Car
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 343 times and lost 657 times.

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