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. Goat
3. Goat
4. Car
5. Car
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 348 times and lost 652 times.

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