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. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Car
7. Car
8. Goat
9. Goat
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 366 times and lost 634 times.

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