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

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Car
7. Goat
8. Goat
9. Car
10. Goat
11. Car
12. Goat
13. Goat
14. Goat
15. Car
16. Car
17. Goat
18. Goat
19. Goat
20. Car
21. Goat
22. Goat
23. Goat
24. Car
25. Goat
26. Goat
27. Car
28. Goat
29. Car
30. Car
31. Goat
32. Car
33. Car
34. Car
35. Car
36. Goat
37. Car
38. Car
39. Goat
40. Goat
41. Goat
42. Car
43. Goat
44. Goat
45. Car
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Goat
58. Car
59. Car
60. Car
61. Goat
62. Car
63. Car
64. Goat
65. Goat
66. Car
67. Car
68. Goat
69. Car
70. Goat
71. Car
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Car
78. Goat
79. Goat
80. Goat
81. Goat
82. Goat
83. Goat
84. Goat
85. Goat
86. Car
87. Goat
88. Goat
89. Car
90. Car
91. Goat
92. Goat
93. Goat
94. Car
95. Car
96. Goat
97. Car
98. Car
99. Goat
100. Car
101. Goat
102. Goat
103. Car
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Car
110. Goat
111. Goat
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. Car
128. Goat
129. Goat
130. Goat
131. Car
132. Goat
133. Car
134. Goat
135. Goat
136. Goat
137. Car
138. Car
139. Goat
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Car
147. Goat
148. Goat
149. Car
150. Goat
151. Car
152. Car
153. Goat
154. Goat
155. Goat
156. Car
157. Car
158. Car
159. Goat
160. Goat
161. Car
162. Goat
163. Goat
164. Goat
165. Goat
166. Car
167. Goat
168. Goat
169. Car
170. Car
171. Goat
172. Goat
173. Car
174. Goat
175. Car
176. Goat
177. Car
178. Car
179. Goat
180. Goat
181. Goat
182. Goat
183. Car
184. Car
185. Goat
186. Goat
187. Goat
188. Goat
189. Goat
190. Car
191. Car
192. Car
193. Goat
194. Car
195. Car
196. Goat
197. Goat
198. Car
199. Car
200. Car
201. Car
202. Car
203. Goat
204. Goat
205. Car
206. Car
207. Goat
208. Goat
209. Car
210. Goat
211. Goat
212. Goat
213. Car
214. Goat
215. Goat
216. Car
217. Goat
218. Goat
219. Goat
220. Goat
221. Car
222. Car
223. Car
224. Goat
225. Car
226. Goat
227. Goat
228. Car
229. Car
230. Goat
231. Goat
232. Goat
233. Car
234. Car
235. Car
236. Car
237. Car
238. Goat
239. Car
240. Car
241. Goat
242. Car
243. Goat
244. Car
245. Car
246. Car
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Car
253. Goat
254. Car
255. Car
256. Car
257. Goat
258. Goat
259. Goat
260. Goat
261. Goat
262. Goat
263. Car
264. Car
265. Goat
266. Goat
267. Goat
268. Car
269. Car
270. Car
271. Car
272. Car
273. Goat
274. Car
275. Car
276. Goat
277. Goat
278. Goat
279. Goat
280. Goat
281. Goat
282. Car
283. Goat
284. Car
285. Goat
286. Car
287. Car
288. Goat
289. Car
290. Goat
291. Goat
292. Car
293. Car
294. Goat
295. Car
296. Car
297. Goat
298. Goat
299. Car
300. Goat
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Goat
307. Car
308. Car
309. Car
310. Goat
311. Car
312. Goat
313. Goat
314. Car
315. Goat
316. Goat
317. Car
318. Car
319. Car
320. Goat
321. Goat
322. Car
323. Goat
324. Goat
325. Goat
326. Goat
327. Goat
328. Car
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Car
335. Car
336. Goat
337. Car
338. Car
339. Goat
340. Goat
341. Goat
342. Car
343. Goat
344. Goat
345. Goat
346. Car
347. Car
348. Goat
349. Car
350. Goat
351. Goat
352. Goat
353. Car
354. Goat
355. Car
356. Goat
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Car
365. Car
366. Car
367. Goat
368. Goat
369. Car
370. Goat
371. Car
372. Car
373. Goat
374. Car
375. Goat
376. Car
377. Goat
378. Goat
379. Car
380. Goat
381. Goat
382. Goat
383. Goat
384. Goat
385. Car
386. Goat
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Car
393. Car
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Goat
401. Goat
402. Goat
403. Car
404. Goat
405. Goat
406. Car
407. Car
408. Goat
409. Car
410. Car
411. Car
412. Goat
413. Goat
414. Car
415. Car
416. Goat
417. Goat
418. Goat
419. Goat
420. Car
421. Car
422. Goat
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Car
429. Goat
430. Goat
431. Car
432. Goat
433. Car
434. Car
435. Goat
436. Goat
437. Goat
438. Car
439. Car
440. Goat
441. Goat
442. Car
443. Car
444. Goat
445. Car
446. Car
447. Goat
448. Goat
449. Goat
450. Goat
451. Goat
452. Goat
453. Car
454. Car
455. Goat
456. Goat
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Goat
463. Goat
464. Goat
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Goat
471. Goat
472. Car
473. Goat
474. Goat
475. Car
476. Car
477. Goat
478. Goat
479. Car
480. Goat
481. Goat
482. Car
483. Car
484. Goat
485. Goat
486. Car
487. Goat
488. Car
489. Goat
490. Goat
491. Goat
492. Goat
493. Goat
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Car
501. Goat
502. Car
503. Goat
504. Car
505. Car
506. Car
507. Goat
508. Goat
509. Car
510. Goat
511. Goat
512. Goat
513. Goat
514. Goat
515. Car
516. Car
517. Goat
518. Goat
519. Goat
520. Goat
521. Car
522. Goat
523. Goat
524. Car
525. Goat
526. Goat
527. Car
528. Car
529. Car
530. Goat
531. Car
532. Goat
533. Car
534. Goat
535. Goat
536. Car
537. Goat
538. Goat
539. Goat
540. Goat
541. Car
542. Goat
543. Car
544. Goat
545. Car
546. Car
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Car
553. Goat
554. Goat
555. Goat
556. Goat
557. Car
558. Goat
559. Car
560. Car
561. Goat
562. Car
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Goat
570. Goat
571. Car
572. Goat
573. Car
574. Goat
575. Car
576. Car
577. Goat
578. Goat
579. Car
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Car
587. Goat
588. Car
589. Goat
590. Car
591. Car
592. Car
593. Goat
594. Goat
595. Car
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Car
602. Car
603. Goat
604. Car
605. Goat
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Car
614. Car
615. Goat
616. Car
617. Car
618. Car
619. Car
620. Car
621. Car
622. Car
623. Goat
624. Goat
625. Car
626. Car
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Car
634. Goat
635. Car
636. Goat
637. Goat
638. Goat
639. Car
640. Car
641. Goat
642. Car
643. Goat
644. Goat
645. Goat
646. Goat
647. Goat
648. Car
649. Car
650. Goat
651. Car
652. Car
653. Car
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Goat
660. Goat
661. Car
662. Goat
663. Goat
664. Goat
665. Goat
666. Car
667. Goat
668. Car
669. Goat
670. Goat
671. Goat
672. Car
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Goat
680. Goat
681. Car
682. Car
683. Goat
684. Car
685. Goat
686. Goat
687. Goat
688. Goat
689. Goat
690. Car
691. Goat
692. Goat
693. Goat
694. Car
695. Goat
696. Goat
697. Car
698. Car
699. Goat
700. Goat
701. Goat
702. Car
703. Goat
704. Goat
705. Car
706. Car
707. Goat
708. Goat
709. Goat
710. Goat
711. Goat
712. Goat
713. Car
714. Car
715. Goat
716. Goat
717. Goat
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Goat
727. Car
728. Goat
729. Car
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Goat
738. Goat
739. Car
740. Goat
741. Goat
742. Car
743. Goat
744. Goat
745. Goat
746. Goat
747. Car
748. Car
749. Goat
750. Goat
751. Goat
752. Goat
753. Goat
754. Car
755. Goat
756. Car
757. Goat
758. Car
759. Car
760. Goat
761. Goat
762. Car
763. Goat
764. Car
765. Goat
766. Goat
767. Goat
768. Goat
769. Goat
770. Goat
771. Goat
772. Goat
773. Car
774. Goat
775. Goat
776. Car
777. Car
778. Goat
779. Goat
780. Goat
781. Goat
782. Goat
783. Car
784. Goat
785. Car
786. Goat
787. Goat
788. Goat
789. Car
790. Car
791. Goat
792. Goat
793. Goat
794. Goat
795. Goat
796. Goat
797. Goat
798. Car
799. Goat
800. Goat
801. Car
802. Goat
803. Car
804. Goat
805. Goat
806. Goat
807. Goat
808. Car
809. Goat
810. Goat
811. Goat
812. Car
813. Car
814. Car
815. Goat
816. Car
817. Car
818. Goat
819. Car
820. Goat
821. Goat
822. Car
823. Goat
824. Goat
825. Goat
826. Goat
827. Car
828. Goat
829. Car
830. Car
831. Car
832. Car
833. Car
834. Goat
835. Goat
836. Goat
837. Goat
838. Goat
839. Goat
840. Goat
841. Goat
842. Goat
843. Goat
844. Goat
845. Goat
846. Goat
847. Car
848. Goat
849. Goat
850. Car
851. Car
852. Goat
853. Goat
854. Car
855. Car
856. Goat
857. Car
858. Car
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Car
865. Car
866. Goat
867. Car
868. Goat
869. Goat
870. Car
871. Goat
872. Goat
873. Car
874. Goat
875. Goat
876. Car
877. Car
878. Car
879. Goat
880. Goat
881. Goat
882. Car
883. Goat
884. Goat
885. Car
886. Goat
887. Goat
888. Goat
889. Goat
890. Car
891. Goat
892. Goat
893. Car
894. Car
895. Car
896. Car
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Goat
905. Car
906. Car
907. Car
908. Goat
909. Car
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Goat
916. Car
917. Car
918. Goat
919. Goat
920. Goat
921. Goat
922. Car
923. Goat
924. Car
925. Goat
926. Goat
927. Goat
928. Goat
929. Goat
930. Goat
931. Goat
932. Car
933. Car
934. Car
935. Goat
936. Car
937. Car
938. Goat
939. Goat
940. Goat
941. Goat
942. Goat
943. Goat
944. Goat
945. Goat
946. Car
947. Car
948. Car
949. Car
950. Car
951. Goat
952. Goat
953. Goat
954. Car
955. Car
956. Car
957. Goat
958. Goat
959. Goat
960. Goat
961. Car
962. Car
963. Goat
964. Goat
965. Goat
966. Goat
967. Goat
968. Goat
969. Goat
970. Car
971. Goat
972. Goat
973. Goat
974. Car
975. Car
976. Car
977. Goat
978. Goat
979. Car
980. Car
981. Car
982. Car
983. Goat
984. Goat
985. Car
986. Goat
987. Car
988. Car
989. Goat
990. Car
991. Goat
992. Car
993. Car
994. Goat
995. Car
996. Goat
997. Car
998. Goat
999. Goat
1000. Car

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 99 times and lost 901 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;
	}
}