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. Goat
4. Car
5. Goat
6. Car
7. Goat
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 348 times and lost 652 times.

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