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. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
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 355 times and lost 645 times.

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