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. Goat
5. Goat
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 342 times and lost 658 times.

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