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. Car
2. Goat
3. Goat
4. Car
5. Car
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 347 times and lost 653 times.

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