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 3 times and lost 7 times.

1. Goat
2. Goat
3. Car
4. Car
5. Goat
6. Goat
7. Car
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 339 times and lost 661 times.

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