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. Car
2. Goat
3. Goat
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 361 times and lost 639 times.

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