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. 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 315 times and lost 685 times.

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