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 4 times and lost 6 times.

1. Goat
2. Car
3. Car
4. Goat
5. Goat
6. Car
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 328 times and lost 672 times.

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