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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Car
6. Goat
7. Goat
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 332 times and lost 668 times.

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