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. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
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 332 times and lost 668 times.

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