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 2 times and lost 8 times.

1. Car
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
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 351 times and lost 649 times.

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