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. Car
5. Goat
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 348 times and lost 652 times.

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