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. Goat
3. Goat
4. Goat
5. Car
6. Car
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 324 times and lost 676 times.

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