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