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. Car
4. Goat
5. Car
6. Car
7. Car
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 311 times and lost 689 times.

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