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. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
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 334 times and lost 666 times.

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