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. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
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 343 times and lost 657 times.

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