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. Car
4. Goat
5. Goat
6. Car
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 329 times and lost 671 times.

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