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. Goat
4. Goat
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 326 times and lost 674 times.

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