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. 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 347 times and lost 653 times.

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