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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
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 361 times and lost 639 times.

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