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. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Car
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 337 times and lost 663 times.

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