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. Goat
2. Goat
3. Goat
4. Car
5. Car
6. Car
7. Goat
8. Goat
9. Goat
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 352 times and lost 648 times.

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