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. Goat
4. Goat
5. Goat
6. Car
7. Car
8. Goat
9. Car
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 332 times and lost 668 times.

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