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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
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 337 times and lost 663 times.

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