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. Goat
5. Car
6. Goat
7. Car
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 337 times and lost 663 times.

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