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 3 times and lost 7 times.

1. Car
2. Goat
3. Car
4. Car
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 320 times and lost 680 times.

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