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. Car
5. Car
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 341 times and lost 659 times.

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