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 5 times and lost 5 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Car
7. Car
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 338 times and lost 662 times.

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