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