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 6 times and lost 4 times.

1. Car
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Car
8. Car
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 343 times and lost 657 times.

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