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. Car
4. Car
5. Goat
6. Goat
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 324 times and lost 676 times.

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