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 3 times and lost 7 times.

1. Car
2. Goat
3. Car
4. Goat
5. Goat
6. Car
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 357 times and lost 643 times.

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