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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
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 331 times and lost 669 times.

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