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 6 times and lost 4 times.

1. Car
2. Car
3. Car
4. Car
5. Car
6. Goat
7. Goat
8. Goat
9. Car
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 330 times and lost 670 times.

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