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. Car
5. Car
6. Goat
7. Goat
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 330 times and lost 670 times.

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