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. 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 314 times and lost 686 times.

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