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 5 times and lost 5 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Car
7. Goat
8. Car
9. Goat
10. Car

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 331 times and lost 669 times.

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