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