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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
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 287 times and lost 713 times.

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