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. Car
4. Goat
5. Goat
6. Goat
7. Car
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 357 times and lost 643 times.

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