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. Car
4. Goat
5. Car
6. Car
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 340 times and lost 660 times.

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