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. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Car
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 336 times and lost 664 times.

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