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. Goat
4. Car
5. Goat
6. Goat
7. Car
8. Goat
9. Car
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 347 times and lost 653 times.

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