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 5 times and lost 5 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Car
9. Car
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 313 times and lost 687 times.

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