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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
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 360 times and lost 640 times.

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