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. Car
4. Goat
5. Goat
6. Goat
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 340 times and lost 660 times.

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