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. Car
3. Car
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 323 times and lost 677 times.

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