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. Car
3. Goat
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 325 times and lost 675 times.

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