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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
10. Car

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 330 times and lost 670 times.

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